HWgs, Rabbltsltons - UZH · Finalll: the rnore diificult step is to go onc periocl aheacl - to...

$: HWgs, Rabbltsltons - UZH · Finalll: the rnore diificult step is to go onc periocl aheacl - to looli at thc next r'r,eek or the next month. \\'e have good rcsults on a tnonlhlv basis$
_ HWgs,RabbltslBY.HRSTNAGROTHEER Scorp

Evolutionary finance is often described as a branch of hehav-

ioral finanre, but University of Zurich Professor Thorsten

Hens insists that evolutionary finance stands alone. "lfyou

think in three dimensions," he says, 'then there are three

things: traditional finance, behavioral finance, and evolution-

ary finance."

Evolutionary finance uses evolutionary dynamics, such

as mutation and selection, to study how trading strategies

and financial innovations evolve. Why does it get less atten-

tion than behaviorat finance? For one thing it's fairty new

Itons

and probabty fewer than 5o people are working in the field.

For another, the math is extremely sophisticated.

Atthough it alt may sound too theoretical, Hens has

applied his behavioral and evolutionary research to private

banking and asset management and claims excellent results.

But there's no chance he'll be going solo as a hedge

fund manager anytime soon: "l don't do the day-to-day busi-

certainlyness because it's not my ski[[," he says. But he ,

relishss his rote as a consuttant in Europe, and Hens seems

to be making a difference.

38 C F A M A G A Z I N E / M A R C H - A P R I L z o o 6

Cri t ics have comptained that behaviora l f inance fa i ls to s in-gle out market inefficiencies in advance, providing ontyafter-the-fact explanations. ls there any work in progressthat might change th is?

I l-ropc so. bLrl orrc shoulcl ncver lorgct that thc so-calleil nullhvpothcsi: - thc thitrg that has to be bc2lten - is that 1.ouci l l l l l () t Pr-ecl ict lhe markct iu l lnr- n.ar: Givcn t l-rat. I thinl<t lre r-c i : s()mc progress, becausc there urc specif ic things thatrrcturr l l r clo r i or l<.

. l he rc i i rc some bchar, ioral he dge lunds, for e xample .

Lhat plav verv specif ic stralegics. ancl lher- generate t:xcessre turns. For exantplc. al f 'uLlcr & Thalcr Asse t I , Ianagernentthel ' plav on the eamings-surprise e l lect. in lr 'hich the marketreacts rnore to ueg:rt i \-c surprises than to posit iVc :urpriscs.

There are othcr- r 'crv spccif ic things. , \ col lcague of r-nine .Bi l l Zicmba in (-anacla. is plaving on u.har is cal leci probabil i -ty u'e ighting - that people czrre rnore about srnal l probabil i -t ics relal ive to r ihat ther. shoulcl. So he plar.s on u.hat iscal lecl thc lnvori te long-shot bras or-r stock options, ancl het l t r r ' r i l r t t r r t t t r ' t : s l t t l l i .

What are some addit ionaI practicaI appl icat ions ofbehavioraI f inance?

I think one shoulcl make a dlst inct ion benveen assel rnanage-nrent ancl pri \ ,atc banking. Assel rnanagernent has to do rvithunderstanding the market as a courplex s).stem, rvhich I do inevolut ionarv I inance. Behaviortr l I inancc is more tai lorcd tcrthc intuit l r-c mistakes that vou make n'hen investrng. ancl themost interesting area of appl icat ion is private bankir.rg.

Sr,r'itzerltrncl is a big place for private btlnking, right?So a lot of people in the rvorld gi i ,e their rnonev to banks inS*' i tzerlancl. Then thev h:rvc to f ind a lvar.to al locate thernone\; not in the Su.rss rnar-ket but in the rvorlci market.

Trprcal mistakes are thatt the1, don't \\,ant to clo a lot ofplanning. This is matchecl bv the behavioral concept ofl-rr-pcrbol ic discourrt ing. -"r 'here r-ou ah'navs think i t 's bette r tostart vour cl ie t tomorrot ' . so \. t)Lr keep postponing ar-rcl post-poning. People don t \vant to läce the init ial cost of thewealth nanagement process. That is t l -rc f i rst rnajor mistake .

The sccond mistake is thcre is a lot of misfranring ol ' t l -rcsituation. He re . be harvroräl f inance is useful becaLrsc rt c; ln Llseb:r lancc shee ts ancl personal asset,/ l iabi l i t l - ntünagerne nt tor c \ ( i l l l ( ) r l i ( r l \ l l r c t t t i r l r l r r r i r r g .

The thircl thing is cl iversif icat ion. There are so manr. r.r-t is-takes that r-ou nrrkc uhcrt r-ou are nai\.c ancl r,oLr tn- to cl ir-cr-si lr- , i rncl behar. ioral f inancc heips to poinl or-rt those mrstal ics.

Final l l , 1' tcople l iavc cl i l l icultr . holcl ing to a strateg\r.lher cl ' rangc t l-reir tnincl er.cn clav or evel\ 'u 'eek. and t l-re1-b.r. ical lv go south in the rnarl ie t bectruse t l-rer- don't l 'ol lou,ir \ tr ' . rr f{ \ ' thrt has been proven to sr,t i t t l ' re ir r isk abi l tn,andthc i r r i :1 . p rc lc rcnces .

What i s your ro le in a t t o f th is?

We harc hclpecl Sn' iss banks l i l<e (-recl i t Suisse irr.rd DeutschcBank Asset \ lanrigcnrcnt clo a structLlrcd uealth managelnentproccss. r,r-hich rs brrsccl or.r bclravioral f inancc. Tlrcv gothrough the proccss nr th c l i cn ts l ronr one s te l l to the ncx t .

1-hev point out the col l trnon pit fal ls ancl shor,r ' cl ients horvto avoicl thent.

I t .s quite a success stor\t espccial l l lor Creci i t Suisse. Inprir-ate banl<ing in Srvitzerland nlonc. (-redit Suisse has some-thing l ike 100.000 cl ients, so vou need a structured process;othe ru'ise . it is totallr' arbitrarl. ar-rcl thc cuslomcrs u,,ondcru'hat thc1. are ptrvir.rg lbr.

Where does evolut ionary f inance fatt within the spectrumof behavioraI f inance?

Mr. point of vier,r' is vor,r shoulcl clistingursh belu.een tra-dit ional. bchar, ioraL. and evolut ionarr. f inance . Mar.be I amtoo str ict on thrs. but bel 'rar, ioral { inance is tarlored to incl ivici-ual rr istakes ancl nol so much to the markels.

Evolut ionarv f inance. on the other hand, is r-rseful forasset rnanagcnrcnt because i t takcs an integratecl vieu. of inter-actior-rs of strategics, not onlv ir-rcl ir-rduals but also inst i tut iot-rs- heclge [unds ancl clelegatecl asset rnanage ment. Ei 'olutronar_vfinance looks art intcr:rct ing strategies and not at incl ivicluals.

Why does evolut ionary f inance not focus on individuatinvestors?

Because the logic of t l ' re rnarket is that indlr- iclual investorsclon't matter. l t .s ps1'cl-rological lrcl i f f icult because vou want tosce vourse l f in the rnoclel. br,rt for the marl<et. i t onlv mattersr'r.hrch strategv is follor,r,ecl br. horv much capital.

I [ r 'ou ]ook at the moclels. r,r 'hat vou uncover pre rry soonis tl.rat asset prlces are not cletern'rinecl br. individuals: thel-are determined bv casl-r I l txr, betu,een the strategies. In theDaru'rnian vreul rvhat is going on in the l 'orest is not deter-mined l t1' the indir idual fox ancl the rabbit but bv the relat iveproport ion ol foxes ancl rabbits.

Can evolut ionary f inance be used to complement tradit ion-aI asset management or does i t take i ts place?

It 's a qr-rest ion o[ u, 'hat vou think tracl i t ional asset manage-ment is. i f Iou thlrk tracl i t iorral asset managen'rent is clorngmeaur-r,ariance calculus and looking at r isk factors, as in thecapital asset pricing moclel. I m prettl. sr-rre 1t r.vill take itspltrcc because the C,\PNI ancl mean-r 'ariance theorv are basedon cross-scctions of returns. But cvolut ionarr, f inance is baseclrun t ir lc series - i t ! basecl on clr 'namical svslems. I f you rvantto ln\,est, rn order to get returns. r-ou hat 'e to understand thed r n a r n i c u l i \ 5 l r ' l l l \ ( ) u i r \ . a ( l i n g i n .

There was a lot of information about behavioraI f inance inthe l i terature, but not much about evolut ionarv f inance.Why is that?

Thc rnain reason is because people think there is tradit lonalf inancc ancl then the re is "son.rething else. ' People clontunclerstand the diffcrence betr,r,een er,olutionan' finar-rce andbchavroral f inance bcciruse thcv alreaclv l-rale di l f icultv uncler-standing the dist inct ion betu,een tradit ional ancl behai ' ioral.So. manv of the results fmm evolut ionan- f inance get suckeclu p i t t h t ' l r a r i , r t ' n l h n r n c . .

M1, second guess is that t l ic mirthcmatics arc not onlvnet, bnt ther., are also hard. I 'm not a mirthematician mvself.

M A R C H . A P R I L 2 o o 6C F A M A G A Z I N E I

I l i 'ork lvi th highlv trained co-authors: a Russtan n'rathemati-

cian and another mathematician based in Englancl. So y'r ' r t t

l l e c d \ L ) n l c i n \ ( \ l l i l ( n l i t l l ( r i l .

Would you expect that more people wit l begin to fol low

evolut ionary f inance as i t becomes better known?

l 'm prettv surc about this. Whe n u'e ir-rvi tecl Daniel

Kahnneman. r.vinner of the 20112 Nobel Prize for behavioral

f inance. to gn-e a presentation herc art the Ulr iversitv ol

Zurich, rr student askecl l-r im about er,olut ionan' f inance , and

he saicl t l iat l t is the luture. , \nd I 'm prctt\- sure that 1-re

kntxvs. The rest of us har,e to do sornething clse , becaruse r,r,e

al l rvant the Nobcl Prize I laughsl :rncl this does seem to be

the futurc for asset rnanage ment.

How does evolut ionary f inance apply to asset management?

First, ,vou need to iclenti f i ' the sct of strategie s I r 'vas Lalking

about. Tl-r:r ts prctt) ' easv because i f vou are an rnvestment

banker or a heclgc älrcl rnanager, )'ou ma): have triecl nanv

such strategies. So you make a l ist of such strategics: rct ivc

ancl passivc. cl ist inguished by' t i rne horizons.

What were the conclusions of your research?

It does not naivel) 'depencl on return. That $.as the f irst gl less- that n-hcr-r rr 5tratcg\ has a high re tuln, everr-bodr,. jumps

on the high return. Rut it clepencls lnore on the rank. When

a strategv is ranked frrst. it gets more inflovns. Ultin-ratelr'.

inr.estors want to cl ir , ide their rrroney across various ol those

strategies, and then t l-rev take the best-ranked strategl i

The f irst-rankecl strategt 'r .r , i l l get more nlon(' \- ncrt pcri

ocl. Ancl il'r'ou understancl rvhat tl're fund is doing - end thi:

is knorvn because if thel' r,r,irnt to atlrlrct mone)', thcn thcl-

have to cl isclose in a se nse what they' are cloing - then you

can make predictions on a meclium-tenn horizon.

Back to the forcst analogr': il ri'e have so man), fbxes ancl

suclclenlv the rabbits die out. we can cio sort o[ a predict ion.

So last \-ear u'as a good r.ear Ibr the Su.iss market, not so

rnuch lbr thc US lnarl<el. ancl ther-r )'ou can irnagine ll'here

the flou's go this ).ear. When vou see the interaction of the

strategies. vou can guess i t . C)[ course. 1'ou need a iot of

econometncs in orcler to f incl t l -re correct values. but lotrcan make some educated guesses.

There are some success stories. The Duke ol l- iechtenstein

Ther-r rvhat'.s n'rore clilficult rs 1'ou haYe to g(rt some nrun-

bcrs lbr the rel: i t i r ,e import irncc of those strategies. Fortunate-

11,. u'e have somc goocl data noll' lrorn State Street on u.hich

traclcs har.e been done bl ntich inr,estors. so \ve can classil,i.'

the strategies h,v the lnr-estors and the n-ealth.- l 'hen,

r 'ou tr1' to czrhbrate the rnodel so that i t l i ts to

rvhat vou har,'e observecl. Finalll: the rnore diificult step is to

go onc periocl aheacl - to lool i at thc next r ' r ,eek or the next

month. \ \ 'e have good rcsults on a tnonlhlv basis but not so

much on a u'eeklv basis.

So you ha\.c strategies. r 'ou have the n'ealth at anv point

in tirne, and tl'ren vori havc to figure out r.r'hat 1'ou call the

u'ealth-f lou, Iunction: u,hen cloes a certain strateg)' att lact

more r'r.ealth than some other stratcgr'?

This is quite u. 'el l knorvn, ancl i t .s quite robust, actr.ral l i r

We cnn obsen'e this r'r.ith hedge funcls or mutual funds. for

example . \\ie have sone nicc criteria to sce r'r'hen one strateg\,

oI a l-reclge fr-rncl attracts r'r.ealth ancl lvhen a certain strate gv

clr ies out. We cl id i t in t}-re laboratorv as r.r,cl l to look al other

florv fur.rctions - the), look cluite sirnilar. pointing at a"unir.ersal lar,r"' almost like those hnor'r.n in ph1'sics.

has a btrnk cal led LGT (Liechtenstein Global - lrust).

\ \ 'e

helpeci create a fund lbr him basecl on thest: evohttionrrr'1-

principles. For fir"e vears no\\'', it has r,r'orked great.

How does evolut ionary f inance relate to genetic algori thms?

A genetic algori thm is one stratcgv to solr,e complicatecl o1'rt i -

mization problerns. There has been some strand of the l i tera-

Hollancl ancl others at the Santa Fe lnstitnte - 11'111:r

thought mavbe the mean-r.etr iance calculus is 1oo simplc and

r.r,c should clo some more sophlst icatecl things. l i l ie genctrc

algori thms, in order to f ind the best investment stretcgic:.

But tlie problern r.r'ith this is, as u-itl-r traditiontrl or behavioral

finance , it'.s onl,v a partial vigu,; it-s onlr, focusing on one class

ol strategies now generated bv genetic algori thms.

Evolut ionarl ' f jnance takes rnto account those strategies.

and thel' don't al"vavs do very rve1l. The1. do better than

mcan-variance . u'hich we can obsen-e arnd also prove mathe-

matical l) l but i t ' .s onlv one ingredient in a pool of stratcgrt ' :

i t t t c r a t l i t t g r r i t h r r l l t t ' r s t r a l c g i r ' \ .

C t A M A G A Z I N E / M A R C H - A P R I L z o o 6

Working from an evolut ionary perspective, MIT's

Andrew Lo has said that " innovation is the kev to survival."

Would you agree?

I t l r ink rt clepends a bit on the t ine horizon. I f r .ou are a hedge

Irr n. l rrrrrugcr - i f vou \\rant to generate return over a short-t , ' r nr irorizon- ) 'ou have to bc r.erf innovative because i t mavju:t l iappen that vour strateg)' is coverecl bv others so that

er crvbocl l , jumps on i t and then the excess returns go awalBut r, 'ou should also keep in mind that rf 1'ou are a dif fer-

ent animal. r 'ou can cLo verv cl i f lerent strtr tegies and thel 'r .r . , i l laln'ays u.ork. For example . Warren Bulfett is doing srrnpiethings. ancl tl-rev alr'vavs r.vork. This is again similar to el'olu-t ionarv brologr,. Dicl 1'ou knorv that scorpions have been thelr,av thcr- are for morc than 400 milllon vcars?

-l-he,v never

changcl thcl ' are just goocl at r,r .hat the1, clo.

Why has the role of delegated investing largely beenignored in f inance?

ln the starndarcl theon-, therc is onh' onc pe rson rvho decicleson cr,ervthing as a representative agent. I t cannot bui ld inclelcgtrtecl asset lnanagement, because 1,ou neecl at least one

',vho clelegatcs ancl one * 'ho cloes the management. I thinl<this rnethodologf is total nonscnse. because vou have to stuch'

t l-re inte r:rct ions of rrarious decisions that are tzrken. r\ singlercprcscntative agent just doesn't f i t .

l1 ror-r look at what pcople in phvsics or bio1og1. aredoing. tl-rcr- u.ould sav u'hat r.r,e are doing are not reallv "mod-

els. ' Thcr' l -oulcl sav i t ls a Nlicker, Mousc thing. Here at t1-reUniversin ol Zuricl-r. thcr- havc 20 rcsearchers rvrth r.arious

backgrounds doing or.re rroclel to stuclv a r,cr)- specihc t1'pe of

c:rncer that you get rn vour bel l l : In f lnance , rvc clo a model(rn oLlr o'"r,n ancl u'e maybe have ln'o stuclents he$ing r-rs, sctrt : l r()t i r coordinated elfbrt as in thc natural sciences.

I think u.'e har-e to rmprove on this. W'e neecl to clo moreol): ! ' r \ i i t i ()ns and computer simulat ior-rs in orcler to get good

uroclcl: . lhc [ irrancial market is even more complex than acanccr . th ( rL lg i r thc mather lä t i cs i s the same. That ' s wh) . I metrvith coi lc.rgLrt: lnrnt the ctrncer rescarch. because thev are alscrusing ranclon clr ulrnrict i l svslerns - in this case , clescl ibingthe relat ive spcccl ol gros'th of I ' realthi ancl cancerous cel ls.

How often do anomalies found in the laboratory end upbeing exploitable in real- l i fe markets?

I onll' knovr' the trvo or three successful behar,ioral heclgelunds rnentionecl earl ier. I knor.v others rvho har-e trted to godirect lv from the laboraton, to the market. but thel ' basical l i ,rvasted their monel ' . The strength of laboraton' rcsearch isthat 1,ou can isolate the aspects vou are interestccl in. But areal market is just too complex comparecl .ur.rth r,vhat ]-ou aredorng in the laboratonr

I lvor-rld be r,r.ar)' of betting mv rnone), on a partialaspect lbund in :r laboraton. experiment, becausc there mar,be a clozen other aspects going exactl ,v the other rvay around.l'm sal,ing this o,en though l knou, rnr,, colleagues r,vill hateme for saving i t , because i t 's so nice to sa1,. ' ' t -ook. lve havefound thls in the laborat66r, and nor,v lve'll go to \\hll Street."

fiaughs] /

Christina Grothttr is a contributing e ditor ontl cu't cditorialtonsultcmt fo CFA Magazine .

Why focus on one fietd when you can do two? Thorsten

Hens, a professor at the Institute for Empirical Research

in Economics at the University of Zurich, currently ispursuing research in both behavioral and evolutionary

finance (3o papers and counting).

Hens also serves as an adjunct professor in the

Norwegian Business School's department of finance in

Bergen, Norway. At the National Center ofCompetencein Research, he is the scientific director of Financial

Valuation and Risk Management (FINRISK), works as aCenter for Economic Policy Research fellow, and collabo-

rates on a European Science Foundation proiect.

Having earned his PhD from Bonn University(Germany) in t992, Hens has held positions at several

universities in Germany, in addition to work in Paris

and at Stanford University.

M A R C H - A P R I L 2 o o 6C F A M A G A Z I N E

Journal of Mathematical Economics 41 (2005) 1–5

Evolutionary finance: introduction to thespecial issue�

Thorsten Hensa, ∗, Klaus Reiner Schenk-Hoppea, b

a Institute for Empirical Research in Economics, University of Zurich, Bl¨umlisalpstrasse 10,8006 Zurich, Switzerland

b Institute of Economics, University of Copenhagen, Studiestræde 6, 1455 Copenhagen K, Denmark

Received 22 October 2002; accepted 30 September 2004Available online 19 December 2004

Abstract

This paper introduces the special issue of theJournal of Mathematical Economicson EvolutionaryFinance.© 2004 Elsevier B.V. All rights reserved.

Keywords:Evolutionary finance; Traditional and behavioral finance model; Financial markets

This special issue of theJournal of Mathematical Economicscontains papers at thecutting edge of research inevolutionary finance, a subfield of financial economics. Evo-lutionary finance provides a synthesis between the traditional and the behavioral financepoint of view.

Recent empirical and experimental work has quite successfully challenged the traditionalview of efficient markets and the long-sustained belief in market rationality, see for examplethe excellent surveys byCampbell (2000)andHirshleifer (2001). Indeed a new paradigm

� The starting point of this special issue is marked by the conference “Evolutionary Finance” held at the SwissExchange (SWX) in Zurich, Switzerland, in 2002. We are grateful to Doyne Farmer from the Santa Fe Institutefor co-organizing and to Richard Meier for sponsoring the conference. The hospitality of the SWX contributed alot to the highly productive atmosphere of the meeting.

∗ Corresponding author. Tel.: +41 1634 3706; fax: +41 1634 4907.E-mail addresses:[email protected] (T. Hens), [email protected] (K.R. Schenk-Hoppe).

0304-4068/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2004.09.001

2 T. Hens, K.R. Schenk-Hopp´e / Journal of Mathematical Economics 41 (2005) 1–5

based on behavioral models of decision under risk and uncertainty is beginning to crowd outthe traditional view based on complete rationality of all market participants. The traditionaland the behavioral finance model however share one important feature: They are bothbased on the notion of a representative agent—though this mythological figure is dresseddifferently. While traditionally he had rational preferences, expectations and beliefs, he iscurrently a prospect theory maximizer, unable to carry out Bayesian updating and likely tofall into framing traps.

Instead evolutionary finance suggests a model of portfolio selection and asset pricedynamics that is explicitly based on the idea of heterogeneity of investors. As a singleindividual has little (or negligible) weight, evolutionary finance suggests to think of thefinancial market not in terms of individuals but in terms of strategies. For the market itis irrelevant who is investing according to, say, P/E-ratios. The only thing that matters ishow much money is invested according to such a criterion. The investors’ strategies mayresult from rational maximization of some intertemporal expected utility function, simpleheuristics, behavioral finance or principal-agent models describing incentive problems ininstitutions. For evolutionary finance it is irrelevanthowstrategies are generated because itanalyzes theirperformanceonce they are in the market.

Evolutionary finance is descriptive and normative as well, answering which set of strate-gies one would expect to be present in a market and how to find the best response to anysuch market. The first observation is that there is nothing like “the best strategy” because theperformance of any strategy will depend on all strategies that are in the market. Rationalitytherefore is to be seen as conditional on the market ecology.

Whereas traditional finance—based on optimization and equilibrium—borrows a lotfrom classical mechanics, behavioral finance borrows from psychology. Instead evolution-ary finance borrows from biology, in particular from biological models based on evolution-ary dynamics. The principles of natural selection and mutation, as formulated by CharlesDarwin, are two fruitful analogies that led evolutionary finance surprisingly far until today.While in biology resources like food are fought for, in finance strategies fight for marketcapital. Selection changes the relative weight of strategies in the market and may lead toextinction for some. On the other hand, mutation enriches the market’s ecology (at leasttemporarily).

The evolutionary finance perspective also leads to a revival of the classical Law ofDemand according to which prices are determined by demand and supply. According totraditional finance, prices are exclusively determined by expectations because agents areable to borrow without limits. Behavioral Finance, using the term “limits to arbitrage”,has convincingly argued that unlimited borrowing is impossible so that prices are alsodetermined by demand from noise traders, for example. Evolutionary finance takes thispoint serious and shows how the flow of wealth between strategies is the driving force ofprice formation. Ultimately, evolutionary finance models try to explain how the ecology ofthe market evolves over time, i.e. how the distribution of wealth across strategies as wellas the strategies themselves change over time. This should also result in better predictionmodels for asset prices.

From its very nature, evolutionary finance models are better suited for empirical researchthan those in traditional finance that is based on the notion of expectations. Data on strategiesthat are pursued and the flow of wealth between these strategies can be collected on actual

T. Hens, K.R. Schenk-Hopp´e / Journal of Mathematical Economics 41 (2005) 1–5 3

markets (for example in the mutual fund and the hedge fund sectors), while it is considerablyharder to quantify traditional models by finding data on expectations. This is so becausemost capital is managed by delegation and in this process the principal (the investor) wantsthe agent (the wealth manager) to commit to some strategy in order to simplify monitoringand reduce verifiability problems. Indeed many banks compete for investors’ money byadvertising investment styles or strategies they want to commit to.

While it is too early at the present stage to assess the potential impact of evolutionaryfinance on our understanding of financial markets, it might be encouraging to recall Theo-dosius Dobzhansky’s dictum that “Nothing in biology makes sense except in the light ofevolution”. A similar statement may well be true for the dynamics of financial markets.

This special issue of theJournal of Mathematical Economicscomprises nine papers ontopics within the emerging field of evolutionary finance. The papers aim to explore theabove ideas in consistent and adequate models with the goal to contribute to a better under-standing of the dynamics of financial markets. The collection of papers in this issue reflectsthe diversity of evolutionary approaches in terms of both conceptual and methodologicalaspects.1

On the conceptual level, the reader will notice several different modeling approaches pur-sued in the papers. Temporary and general equilibrium models are considered. Dynamicalsystems theory as well as game-theoretic reasoning is applied. Agents’ behavior originatesfrom expected utility maximization, genetic learning, or is only restricted by being adaptedto the information filtration. Fundamentalists and noise traders also enter the stage. Onthe methodological level, a variety of treatments, analytical, empirical and numerical, havebeen applied by the authors. In the best tradition of theJournal of Mathematical Economicsall papers share a thorough and formal treatment of the issue under consideration.

Brock et al. (2005)analyze in their paper “Evolutionary dynamics in markets with manytrader types” the dynamical behavior of an asset market as the number of buyer and seller’stypes tends to infinity. A large type limit result is established in a rigorous mathematicalfashion. It turns out that the limit dynamics, which provides an approximation of an economywith a large but finite number of traders, is deterministic. A specific asset market model withheterogenous beliefs (inspired byBrock and Hommes (1997, 1998)) is studied in detail.The dynamical properties of the limit dynamics in dependence of structural parameters ofthe economy, which can include different bifurcation scenarios, does generically reflect thedynamics in the original market.

In Thorsten Hens and Klaus Reiner Schenk-Hoppe’s paper “Evolutionary stability ofportfolio rules in incomplete markets”, the endogenous wealth dynamics between differentinvestment strategies is studied. It is shown that among all stationary investment strategiesthere is only one which is not destabilized by the introduction of any “mutant” strategy.This investment rule has an explicit representation: it prescribes to allocate one’s budgetacross assets according to the expected value of their relative dividend payoffs. In the longrun, relative asset prices are thus equal to their relative fundamental values. The main resulthas a game-theoretic interpretation—who wants to play a dominated strategy?

1 It is tempting to remark that this special issue on evolutionary finance is an element of our strategy in competingfor the professions’ attention in a truly heterogenous population of approaches to the understanding of financialmarkets.

4 T. Hens, K.R. Schenk-Hopp´e / Journal of Mathematical Economics 41 (2005) 1–5

In “The asset market game”Al os-Ferrer and Ania (2005)elaborate this game-theoreticperspective in a simple financial market model which was first presented inBlume andEasley (1992). The weight, or frequency, of different investment strategies evolves througha process of imitation based on expected returns (but not according to an endogenous wealthdynamics). The authors also study a proportional imitation rule based on realized returns. Itturns out that a modified Kelly rule is the unique evolutionary stable strategya la Schafferin this game.

Sandroni (2005)’s paper “Market selection when markets are incomplete” aims to gen-eralize selection results bySandroni (2000)andBlume and Easley (2000)to incompletemarkets. To this end a general equilibrium model of an asset market in which consumersmaximize expected discounted utility is studied. The question is analyzed whether the utilityfunction and/or the accuracy of beliefs matter for survival. In complete markets it is knownthat only the latter matters. Sandroni’s results show that in incomplete markets a conceptof entropy relative to the market’s belief is needed.

The long-run behavior of arbitrary incomplete markets of short-lived assets is analyzedin “Market selection and survival of investment strategies” byAmir et al. (2005). In thispaper it is proved that asymptotically either the market portfolio is held by all investors orthe investment strategy discovered inHens and Schenk-Hoppe (2005)(the expected valueof the relative dividends rule) will dominate the market. Since every adapted investmentstrategy is permitted, this result is in particular true in a general equilibrium asset marketmodel.

Follmer et al. (2005)in their paper “Equilibria in financial markets with heterogeneousagents: a probabilistic perspective” analyze financial market models in which agents formtheir demand for an asset on the basis of their forecasts of future prices and where theirforecasting rules may change over time, as a result of the influence of other traders. Pricescan exhibit transient behavior when chartists predominate. However, if the probability thatan agent will switch to being a “chartist” is not too high then the process does not explodeand the limit distribution of the price process exists and is unique.

At the heart of the models discussed above lies the long-run performance of investmentstrategies—this has normative appeal. However, the papers also acknowledge that the degreeof risk aversion matters considerably for the evolution of the wealth distribution.Levy (2005)in his paper “Is risk-aversion hereditary?”, attempts to shed more light to what extent ourpreferences are learned or inherited. Empirical observations are shown to provide an upperbound on the role of heredity in determining risk-aversion.

Lux and Schornstein (2005)approach the problem of the explanatory power of heteroge-nous agent models in their thorough complex-adaptive-systems paper “Genetic learning asan explanation of stylized facts of foreign exchange markets”. It turns out that many reg-ularities in actual currency markets can be observed in artificial markets with adaptiveinvestment behavior that stems from genetic learning algorithms.

The importance and effect of different trading protocols on the outcome of market in-teraction is analyzed byBottazzi et al. (2005)in their paper “Institutional architectures andbehavioral ecologies in the dynamics of financial markets”. In particular, they show thatmarket architectures bear a central influence upon the time series properties of market dy-namics. Conversely, the revealed allocative efficiency of different market settings is stronglyinfluenced by the trading behavior of the agents.

T. Hens, K.R. Schenk-Hopp´e / Journal of Mathematical Economics 41 (2005) 1–5 5

It is our hope that this collection of papers, which encompasses several directions ofcurrent developments inEvolutionary Finance, serves the interested reader and stimulatesfurther research in this exciting field.

References

Al os-Ferrer, C., Ania, A.B., 2005. The asset market game. Journal of Mathematical Economics 41, 67–90.Amir, R., Evstigneev, I.V., Hens, T., Schenk-Hoppe, K.R., 2005. Market selection and survival of investment

strategies. Journal of Mathematical Economics 41, 105–122.Blume, L., Easley, D., 1992. Evolution and market behavior. Journal of Economic Theory 58, 9–40.Blume, L., Easley, D., 2000. If you’re so smart, why aren’t you rich? Belief selection in complete and incomplete

markets, mimeo. Department of Economics, Cornell University.Bottazzi, G., Dosi, G., Rebesco, I., 2005. Institutional architectures and behavioral ecologies in the dynamics of

financial markets. Journal of Mathematical Economics 41, 197–228.Brock, W.A., Hommes, C.H., 1997. A rational route to randomness. Econometrica 65, 1059–1095.Brock, W.A., Hommes, C.H., 1998. Heterogenous beliefs and routes to chaos in a simple asset pricing model.

Journal of Economic Dynamics and Control 22, 1235–1274.Brock, W.A., Hommes, C.H., Wagener, F.O.O., 2005. Evolutionary dynamics in markets with many trader types.

Journal of Mathematical Economics 41, 7–42.Campbell, J., 2000. Asset pricing at the millenium. Journal of Finance 55, 1515–1567.Follmer, H., Horst, U., Kirman, A.P., 2005. Equilibria in financial markets with heterogeneous agents: a proba-

bilistic perspective. Journal of Mathematical Economics 41, 123–155.Hens, T., Schenk-Hoppe, K.R., 2005. Evolutionary stability of portfolio rules in incomplete markets. Journal of

Mathematical Economics 41, 43–66.Hirshleifer, D., 2001. Investor psychology and asset pricing. Journal of Finance 56, 1533–1597.Levy, M., 2005. Is risk-aversion hereditary? Journal of Mathematical Economics 41, 157–168.Lux, T., Schornstein, S., 2005. Genetic learning as an explanation of stylized facts of foreign exchange markets.

Journal of Mathematical Economics 41, 169–196.Sandroni, A., 2000. Do markets favor agents able to make accurate predictions? Econometrica 68, 1303–1341.Sandroni, A., 2005. Market selection when markets are incomplete. Journal of Mathematical Economics 41,

91–104.

Survival of the Fittest on Wall Street∗

Thorsten Hens und Klaus Reiner Schenk-Hoppé

1. Introduction

Recent empirical and experimental work has done a lot to undermine thelong sustained belief in market rationality (see e.g. the surveys by Campbell(2000), Hirshleifer (2001) and De Bondt (1999)). These important findingshave initiated a new behavioral paradigm for finance that – according tomany researchers in the field – might replace or at least complement tra-ditional finance. It is currently believed that thinking in terms ofexcessvolatility, irrational exuberance, market riskand loss aversionwill soonsubstitute the cornerstones of traditional finance,mean-variance analysis,arbitrage pricingand theefficient market hypothesis. While the fight of therational and the behavioral finance paradigm is in its decisive stage, weargue that a third paradigm, evolutionary finance, should not be omitted.Thinking in terms ofstrategies, market selectionandmutationseems to bevery appropriate for finance. In this view, for example, a stock market isunderstood as a heterogeneous population of frequently interacting portfo-lio strategies in competition for market capital. Market selection is perhaps

∗ We thank Martin Stalder for the simulation program and Claudio Mazzoni fromBank Leu for providing the DJIA data. Financial support by the national centre ofcompetence in research “Financial Valuation and Risk Management” is gratefullyacknowledged. The national centers in research are managed by the Swiss NationalScience Foundation on behalf of the federal authorities.

424 Thorsten Hens und Klaus Reiner Schenk-Hoppé

most severe in these markets and innovations, respectively mutations, oc-cur frequently.

The aim of our paper is to contribute to a Darwinian theory of portfolioselection. This theory views asset markets as being stratified according tothe portfolio rules that investors use to manage wealth. The building blocksof the model are therefore strategies but not the individual investor, i.e. foreach strategy all wealth being managed by that strategy is added up. This isanalogous to Darwin’s view according to which the species but not the indi-vidual animal counts for evolution. The strategies considered in this paperare the mean-variance rule, the growth-optimal rule, the CAPM rule, naïvediversification, prospect theory based rules and a relative-dividends rule.In our model the impact of any such rule on market prices is proportionalto the amount of wealth managed by the rule.

In a Darwinian model two forces are at work: one reducing the varietyof species and one increasing it. In our model the first such force is theendogenous return process acting as a market selection mechanism thatdetermines the evolution of wealth managed by the portfolio rules. That isto say, if some rule has gained wealth because it has managed to buy lowand to sell high then other rules must have lost an equal amount of wealth.Secondly, any system of portfolio rules that is selected by the market se-lection process is checked for its evolutionary stability, i.e. it is checkedwhether the innovation of a new portfolio rule with very little initial wealthcan grow against the incumbent rule.

The Darwinian theory of asset markets seems to describe very well amodern asset market in which most of the available capital is investedby delegated management. Indeed investors typically choose funds by theportfolio rules, also called “styles,” according to which the money is in-vested. Style consistency appears nowadays to be one of the most impor-tant features in monitoring fund managers.

A long time ago Friedman (1953) and Fama (1965) have already rec-ognized the power of evolutionary ideas in finance. Using these ideas theyargued that the market naturally selects for the rational strategies. As aneffect market selection would lead to market efficiency. This specific out-come of the market selection process could not be sustained in general.For example De Long, Shleifer, Summers and Waldmann (1990) show thatunder specific circumstances noise traders can earn a higher average rateof return than rational arbitrageurs. While this example has been very in-

Survival of the Fittest on Wall Street 425

fluential in the debate for behavioral finance it has some shortcomings thatshould be removed. In particular it is based on the model of overlappinggenerations which implies that market selection has no bite. Indeed in DeLong, Shleifer, Summers and Waldmann (1990)’s example every strategyis renewed with fresh capital in every period. In our model every strat-egy will have to continue investing with the wealth it has generated in theprevious period so that the market selection process has more bite. Stillwe are able to show that seemingly rational strategies, like mean-varianceoptimization, can do very poorly against seemingly irrational strategies,like naïve diversification according to which wealth is distributed equallyamong the investment opportunities.

Some considerable progress in the field of evolutionary finance has beenmade since Friedman (1953) and Fama (1965). This progress was madepossible due to a formalization of evolutionary reasoning based on newdecision models like quantifier systems, for example, using computer sim-ulations and advanced mathematical techniques. Many time series proper-ties of asset prices have found an explanation by evolutionary reasoning(Arthur, Holland, LeBaron, Palmer and Taylor (1997), LeBaron, Arthurand Palmer (1999), Brock and Hommes (1997), and Lux (1994) amongothers). These results have also received very good recognition in practi-tioners’ news letters (Mauboussin (1997)). Our paper contributes to thisevolutionary asset pricing theories by showing that the market selectionprocess studied here can also generate the phenomenon of stochastic timeseries of asset prices that do not converge. The interaction of strategies canlead to endogenous volatility in returns without any convergence of assetprices. While the asset price dynamics in the standard evolutionary assetpricing models is complex and hence not easy to interpret, the explanationof the time series properties of asset prices arising in our model is quitesimple. In our model it is the endogenous change in the wealth shares thatgenerates and amplifies fluctuations in asset prices. A particular examplewith two strategies is given in which for both strategies market prices turnto the disadvantage of a strategy as its market share becomes large. Thusthe more wealth is managed by a strategy, the higher is the growth potentialof the other strategy – eventually leading to a reversed evolution of marketshares.

In the area of portfolio theory the seminal work of Blume and Easley(1992) has laid the foundations for a series of papers (Sandroni (2000),


Blume and Easley (2001), Sciubba (1999), Hens and Schenk-Hoppé(2004), Evstigneev, Hens and Schenk-Hoppé (2002), Amir, Evstigneev,Hens and Schenk-Hoppé (2004), Evstigneev, Hens and Schenk-Hoppé(2003)) developing a variety of evolutionary portfolio models. This theoryprovides a framework in which the market selection hypothesis put forwardby Friedman and Fama can be studied. It turns out that as long as there areexcess returns there still exist strategies that can gain market wealth at theexpense of the existing strategies. Moreover, there is one strategy, the evo-lutionary portfolio rule discovered in Hens and Schenk-Hoppé (2004), thateliminates all excess returns and that cannot be driven out of the market byany other strategy that is adapted to the information revealed by the historyof the states.

While the theoretical papers on evolutionary portfolio selection deriveasymptotic results in idealized markets, the point of this paper is to applythe evolutionary ideas to stocks from the Dow. The portfolio choice consid-ered in our paper is the decision how to allocate wealth among shares yield-ing a dividend as observed in the Dow data. We focus attention on dynamicportfolio strategies. That is to say we view a modern capital market, likethe stocks in the Dow, as a heterogenous population of dynamic portfoliostrategies. These strategies interact repeatedly via the market mechanismand are thereby competing for market capital. Instead of considering alltheoretically possible dynamic portfolio rules we take a more pragmaticpoint of view here and restrict attention tofix-mix rules. A fix-mix ruleholds certain portfolio weights constant for a long period. Hence if marketprices fluctuate a fix-mix rule has to adjust the number of shares it holds soas to keep the proportions of wealth in its portfolio constant. Many insti-tutional investors follow simple fix-mix rules. Some of them because theyhave committed to manage third parties’ money according to a certain assetallocation1, some because they believe that fix-mix is an optimal behavior

1 In many prospects of mutual funds some asset allocation, e.g. 60% technologystock and 40% bricks-and-mortar stocks, is proposed as an optimal investment ruleso that the investors would feel cheated if these proportions fall out of balance. Alsohedge funds for example commit to certain strategies in order to increase credibilityand to reduce monitoring costs.


in volatile markets2 and some because they use trading strategies derivedfrom some clever reasoning like contrarian behavior that in essence are fix-mix rules3. As Evstigneev, Hens and Schenk-Hoppé (2002) have demon-strated in the case of short-lived assets, the simple evolutionary fix-mixrule considered in this paper is not only able to outperform any other sim-ple portfolio strategy but it will also dominate any general portfolio strat-egy given it is adapted to the price process. Hence, even though this paperconsiders the more general case of long-lived assets, it may be argued thatthe restriction to simple fix-mix rules does not restrict the outcome of themarket selection process.

Since from the market selection point of view the market interaction ofthe various portfolio rules is decisive, we cannot simply do an empiricalstudy of the relative performance of fix-mix rules on a given return path.This would ignore the impact one strategy has on its competitors4. Hencewe have to rely on simulations in order to show the would-be performanceof various portfolio rules that are interacting in a market with Dow div-idends. It turns out that the best fix-mix strategy for exogenous returns,the growth optimal portfolio, also called the maximum growth strategy(Hakansson 1970), is no longer the best performing strategy once marketinteraction is taken into account. Our simulations show that in competi-tion with fix-mix rules derived from mean-variance-optimization, maxi-mum growth theory and from behavioral finance the evolutionary financerule discovered in Hens and Schenk-Hoppé (2004) will eventually hold to-tal market wealth. According to this simple rule, hereafter denoted byλ∗,

2 Suppose for example that prices follow a random walk, an often hold assertion,then fix-mix means that on average one buys cheap and sells high. Indeed it can beshown that with idealized returns expected utility maximizers with constant relativerisk aversion will choose fix-mix strategies (see for example Campbell and Viceira(2002)).

3 Following a contrarian behavioral strategy, like that of De Bondt and Thaler(1985) for example, one sells those stocks that have gone up and buys those thatwent down, which is also the main feature of fix-mix rules.

4 Note that we are not claiming that individual traders have a huge impact on marketprices. From an evolutionary point of view the strategy according to which marketcapital is managed is crucial, it is not important whether fund X or fund Y or bothare using this strategy. Hence it may be that a certain commonly used strategy likethe mean-variance-rule has a huge impact while no individual fund has any impact onmarket prices.


the portfolio weights should be proportional to the expected relative divi-dends of the assets. Note that the portfolio weights used by the evolution-ary strategy are based solely on fundamentals, ignoring any price fluctua-tions! As it turns out, in the long run as its wealth share grows, prices willstop fluctuating and settle down on the relative expected dividends of theassets because eventually only the single surviving evolutionary rule willdetermine market prices. This is of course a very strong prediction. Oursimulations show that even though the final proceeds of the evolutionaryprocess are huge (one gathers total market wealth), one might have to waitvery long before this happens. However, even a very modest investor willbe pleased with the growth of the market share of the evolutionary rule.Starting from equal grounds, after 8 periods the evolutionary rule has dou-bled its market share for the first time and it takes another 50 periods todouble it once more. And starting from a market share of only0.1%whichis 1% of the others’ market shares, the first doubling of the evolutionaryrule’s market shares happens after only 4 periods, the second doubling after17 periods and the third doubling after 40 periods and after 50 periods ithas reached 10 times its initial market share. Hence in contrast to the wellknown critique on the maximum growth literature, put forward for exampleby Rubinstein (1991), the convergence of the process is much faster whenprices are endogenous. It should also be noted that every run of the simu-lation looks pretty much the same. Indeed, the variance over the differentruns of the simulations is negligible. Based on 30 runs we found that thevariance of the market shares averaged over all periods is only0.36%.

In passing we would like to mention that for the case of long-lived as-sets considered here, so far the theoretical literature has not been able toprove what our simulations show: the global convergence of the evolu-tionary process towards a situation in which all wealth is managed by theevolutionary ruleλ∗.5 Hence our “application” of the evolutionary portfo-lio theory also hints at new theoretical results. Recently, Evstigneev, Hensand Schenk-Hoppé (2003) have shown that in the set of all adapted strate-giesλ∗ is the unique evolutionary stable strategy. This is to say ifλ∗ holdsall market wealth then every mutant strategy that enters the market with a

5 For the case of short-lived assets, global convergence to the evolutionary rulehas been demonstrated under very general conditions (Amir, Evstigneev, Hens andSchenk-Hoppé 2004).


small fraction of wealth will driven out of the market. Moreover onlyλ∗

has this property, i.e. every strategy different fromλ∗ can be driven out bysome other strategy. This results may help to explain why rational marketslike those in which prices are equal to relative dividends are more stablethan irrational markets in which prices depart from their fundamental val-ues. Also this result shows that if the market selection process convergesthen it has to converge toλ∗, giving some theoretical foundation for thesimulations in whichλ∗ is the single survivor.

In the next section we briefly recall the evolutionary portfolio modelof this paper. Section 3 shows how to apply this model to a market withdividends taken from the Dow. Section 4 presents the results and section 5tries to provide some intuitive explanation of the observed phenomena.Section 6 concludes.

2. An Evolutionary Stock Market Model

We consider a financial market withK ≥ 1 long-lived assetsk = 1, ...,Kin unit supply, each paying an uncertain dividendDk

t ≥ 0 at any period intime t = 0,1, ... . Dividends pay off a perishable consumption good, as inthe seminal paper by Lucas (1978).

Normalizing the price of the consumption good to one in all periods intime, an investor’s wealth in terms of the numeraire is given by

wit+1 =

K

∑k=1

(Dk

t+1 + pkt+1

)θi

t,k (1)

(θit,1, ...,θ

it,K) denotes investori’s portfolio andpk

t is assetk’s price in pe-riod t. They are determined by

θit,k =

λit,k wi

t

pkt

and pkt =

I

∑i=1

λit,k wi

t = λt,k wt (2)

whereλit,k is investori’s budget share assigned to the purchase of asset

k. Prices are determined by equating each asset’s market value with theinvestment in that asset (supply is normalized to one).


Assume all investors consume the same fraction of their wealth in allperiods in time. Denoting the budget share allocated to consumption byλ0 > 0, one has

Dt =K

∑k=1

Dkt = λ0

I

∑i=1

wit = λ0Wt (3)

Then (1) defines an equation for investors’ market sharesr it = wi

t/Wt :

r it+1 =

K

∑k=1

(λ0dk

t+1 +I

∑j=1

λ jt+1,k r j

t+1

)λi

t,k r it

∑Ij=1λ j

t,k r jt

(4)

wheredkt+1 = Dk

t+1/Dt+1 denotes assetk’s relative dividend payoff. It isassumed that at least one asset pays a dividend,Dt+1 > 0. The last equationis linear inrt+1 = (r1

t+1, ..., rIt+1). Its solution is given by

rt+1 = λ0

Id−

[λi

t,krit

λt,krt

]k

i

Λt+1

−1[

K

∑k=1

dkt+1

λit,kr

it

λt,krt

]

i

(5)

where ΛTt+1 = (λT

t+1,1, ...,λTt+1,K) ∈ RI×K denotes the matrix of budget

shares in periodt +1.Equation (5) governs the evolution of market shares for given trading

strategies of investors. It is referred to as themarket selection process.Dividend payoffs are determined by the states of nature revealed up to

and including timet +1. The state of natureωt ∈ S(whereS is a finite set)is governed by a stationary stochastic process. The relative dividenddk

t =dk

t (ωt), where the observed history of states is denoted byωt = (ω0, ...,ωt).A trading strategy is a sequence of budget sharesλi

t = (λ0,λit,1, ...,λ

it,K)

with λ0 +∑Kk=1λi

t,k = 1. λit can depend on all past observations but neither

on current market-clearing prices nor on other investors’ current strategies.The evolution of market shares is well-defined if no bankruptcy occurs

and markets always clear. If short sales are allowed, bankruptcy would beprevalent because equilibrium is only temporary in our approach.

The following conditions, which include the absence of short selling,ensure (5) to be well-defined. Suppose that for allt, λi

t,k ≥ 0 (for all i,k)

and there is an investor withr jt > 0 such thatλ j

t,k > 0 for all k. Then (5) isa well-defined map on the simplex∆I = {r ∈ RI | r i ≥ 0,∑i r

i = 1}. For aproof see Evstigneev, Hens and Schenk-Hoppé (2003, Proposition 1). Eq.


(5) generates a (non-autonomous) random dynamical system on∆I . Forany initial distribution of wealthw0 ∈ RI

+, (5) defines the path of marketshares on the event tree with branchesωt . The initial distribution of marketshares is given by(r i

0)i = (wi0/W0)i .

The wealth of a strategyi in any period in time can be derived from hermarket share and the aggregate wealth, defined by (3), as

wit+1 =

Dt+1(ωt+1)λ0

r it+1 (6)

The further analysis is restricted to the case of simple strategies and i.i.d.dividends. We make the following assumptions.

(B.1) Simple strategies, i.e.λi ∈ ∆K+1 for all i = 1, ..., I andλi0 = λ0.

(B.2) I.i.d. dividend paymentsdkt (ωt) = dk(ωt), for all k = 1, ...,K and

the state of natureωt follows an i.i.d. process.

3. Evolutionary Investment

In this section we derive and motivate an evolutionary investment ruleλ∗

which was first discovered in Hens and Schenk-Hoppé (2004) in a simplermodel. This portfolio rule is the only candidate for a rule that can attractall market wealth. That is to say, supposing the market selection process(5) converges, then the portfolio rule that conquers the whole market has tobe the evolutionary investment ruleλ∗. It is important to point out that inall our simulations convergence of the process was obtained ifλ∗ is amongthe set of strategies. We also give an interpretation of the evolutionary in-vestment ruleλ∗ in terms of the well-known growth optimal portfolio rule(Hakansson 1970). It turns out thatλ∗ is the growth optimal portfolio rulein a population of rules which generates prices equal toλ∗.

To this end we analyze the market selection process close to the one-owns-all states, i.e. we investigate the local dynamics close to the verticesof the simplex of market shares. We make the non-redundancy assumption

(C) Absence of redundant assets, i.e. the matrix of relative dividend pay-ments(dk(s))k=1,...,K

s∈S has full rank.Under this assumption one-owns-all states are the only determinis-

tic steady states of the market selection process (Evstigneev, Hens andSchenk-Hoppé 2003). The local dynamics close to a one-owns-all state is


governed by the linearization of the original dynamics. We give a heuristicderivation here and refer the reader to Evstigneev, Hens and Schenk-Hoppé(2003) for the correct mathematical approach.

Suppose one strategy, say strategyj, owns the market wealth. The in-vestment strategy,λ j then determines prices in this case. One haspk

t =λ j

kWt , andpkt+1 = λ j

kWt+1. Under this assumption, (1) and (2) yield

r it+1 =

K

∑k=1

Dkt+1(ω

t+1)+ pkt+1

Wt+1

λik wi

t

pkt

=K

∑k=1

(Dk

t+1(ωt+1)

Wt+1+λ j

k

)λi

k wit

λ jkWt

=K

∑k=1

(λ0dk(ωt+1)

λik

λ jk

+λik

)r it =

(1−λ0 +λ0

K

∑k=1

dk(ωt+1)λi

k

λ jk

)r it

where (6) implies thatλ0dk(ωt+1)= Dkt+1(ω

t+1)/w jt+1 = Dk

t+1(ωt+1)/Wt+1.

The exponential growth rate of strategyi’s market share atλ j -prices canbe inferred from this equation. It is given by

gλ j (λi) = E ln

[1−λ0 +λ0

K

∑k=1

dk(s)λi

k

λ jk

](7)

whereE denotes expected value with respect to the distribution on the setof states of natureS.

Evstigneev, Hens and Schenk-Hoppé (2003) show the following result.

Theorem 1 The portfolio ruleλ∗, defined by

λ∗k = (1−λ0) Edk(s), k = 1, ...,K (8)

is the only investment strategy that is locally stable against any other port-folio rule. More precisely,gλ∗(λ) < 0 andgλ(λ∗) > 0 for all λ 6= λ∗.

Hence supposing the evolutionary process of wealth converges, it canonly converge toλ∗. The above result assumes that investment strategiesare distinct across investors. How can one analyze the case in which, forinstance, more than one investor adopts theλ∗ strategy? Fortunately, eventhe general case of investors pursuing the same portfolio rule is straight-forward: Since the relative wealth shares of two investors with the sameinvestment rule is fixed over time, it is equivalent to assume that investorswith the same strategy set up a fund with claims equal to their initial share.


The implications on the asset prices are immediate from Theorem 1.According to the strategyλ∗ one has to divide wealth across assets pro-portional to the present expected value of their (relative) future dividendpayoffs. The discounting rate is the inverse of the saving rate1−λ0. If theλ∗ portfolio rule manages all market wealth then all asset prices are givenby this vector of fundamental values.

In the long run only theλ∗ strategy is present in the market. Thus theλ∗-investors hold all wealth and asset prices are given by their fundamentalvalues. Since onlyλ∗-investors survive, all surviving investors hold themarket portfolio.

The following corollary shows the relation of the strategyλ∗ to thegrowth optimal portfolio.

Corollary 1 The portfolio ruleλ∗k = (1−λ0) Edk(s) is the growth optimalinvestment strategy in a population where itself determines the asset prices,i.e.

λ∗ = arg maxµ∈∆K+1:µ0=λ0

E ln

(∑k

dk(s)+λ∗kλ∗k

µk

).

The proof of Corollary 1 is analogously to Theorm 1.Before turning to the application we point out that one implication of

equation (7) is that under-diversification is fatal for investment. Supposingsome strategy does not use all assets it can easily be driven out by anycompletely diversified strategy. In particular the illusionary diversificationrule according to which one puts equal weights on all assets can drive outsophisticated rules based on some optimization criterion like for examplethe mean-variance rule.

Corollary 2 Suppose some incumbent ruleλ j with λ jk = 0 for some asset

k has conquered the market. Then any portfolio ruleλi with λik > 0 for all

k grows againstλ j , i.e. gλ j (λi) > 0. (In fact the growth rate is arbitrarilylarge.)

4. Application to the Dow

In this section we apply the general evolutionary portfolio theory modeloutlined above to dividend data from the Dow Jones Industrial Average.


To this purpose we consider the total dividends paid by 21 stocks from theDow in the years 1981 to 2001 (Appendix A lists the data). Those yearsand stocks have been selected in order to obtain a complete data set. Forother stocks and years, the necessary data were not available. We interpretthe data in terms of our model as follows.

First we assume that the 21 years are 21 realizations of a stochasticdividend process. The data reveal that the total dividends of each stockfollow some growth path. Each datet = 1, ...,21 is thus identified with astates= 1, ...,21. Each row of the matrix in Appendix A collects the totaldividend payoffDk(s) of all assetsk = 1, ...,21 for one realization of theexogenous state of nature. Each column contains the dividend payoff of therespective asset across different states. In the simulation relative dividendsare applied, i.e. according to the model every entry in a particular row isnormalized by the sum of that row.

All portfolio strategies considered have to devote the same proportionof wealth to cash holdings. It is assumed to be1%(i.e. λ0 = 0.01).

We consider two types of portfolio rules. Those based solely on (exoge-nous) dividends and those based on (endogenous) returns. Of the first typeis the behavioral finance rule which Benartzi and Thaler (1998) have calledillusionary diversification

λilluk = (1−λ0)

1K

, k = 1, ...,K.

According to this rule budget shares are set equal for all risky assets. Be-nartzi and Thaler (1998) found that surprisingly many investors use thisnaïve rule. Of the first type is also the evolutionary portfolio rule discov-ered in Hens and Schenk-Hoppé (2004):

λ∗k = (1−λ0) E

(Dk

∑Kj=1D j

)= (1−λ0) ∑

s∈S

psDk(s)

∑Kj=1D j(s)

, k = 1, ...,K.

The evolutionary rule presumes that agents calculate expected values cor-rectly. It is however well known from behavioral finance that actual de-cisions of investors are based on perceived probabilities that may not co-incide with the probabilities governing the relevant stochastic process. Toallow for this behavioral distortion, Tversky and Kahnemann (1992) havesuggested a certain transformation functionα : [0,1] → [0,1] that over-states small probabilities and understates high probabilities. This function


is known as thecumulative prospect theory. We have used the cumulativeprospect theory function as suggested by Tversky and Kahnemann (1992)to create a second behavioral finance rule based on the portfolio ruleλ∗.That is to say the portfolio rule based on cumulative prospect theory isgiven by

λcptk = (1−λ0) ∑

s∈S

α(ps)Dk(s)

∑Kj=1D j(s)

, k = 1, ...,K.

The second type of portfolio rules that we consider are those based onreturns. Since we only want to consider simple portfolio rules, i.e. thosewith time independent budget shares, we have to choose some prices thatremain constant in the computation of the returns. In order not to base thesecond type of portfolio rules on some unreasonable prices we give themthe advantage of allowing them to use the prices that eventually will emergein the evolutionary process. Since we are mainly interested in the long runbehavior of the process, these are the prices that determine the long runreturns. As proven for the case of short-lived assets in Evstigneev, Hensand Schenk-Hoppé (2002), our simulations in the case of long-lived assetsshow that those prices are given byλ∗. Hence we give the return basedportfolio rules the advantage of knowing theλ∗-prices.

One of the most prominent examples of return based portfolio rules isthe mean-variance rule suggested by Markowitz (1952). This rule is cer-tainly one of the cornerstones of traditional finance. The interest rate isset to zero because the price of the consumption good is identical in allperiods. We denote it by

λµ−σ(λ∗).

The mean-variance model has always been criticized for using a riskmeasure, the variance, that has too many undesirable features. For exam-ple, it is well known that mean-variance optimization does not necessarilyagree with first order stochastic dominance. Several alternatives, like semi-variance and Value-at-Risk for example, have been suggested in the courseof this discussion. A recent concept is conditional Value-at-Risk, CVaR,which is one possiblecoherentrisk measure as Artzner, Delbaen, Eber andHeath (1997) have argued. The CVaR takes the expectation of the returnsbelow some quantile of the return distribution. The quantile is usually cho-


sen to the5%-level. Based on this idea and the general assumptions madeabove, we generate the portfolio rule

λCVaR(λ∗).

The third portfolio rule based on returns is the growth optimal portfolio(Hakansson 1970). This portfolio strategy maximizes the expected growthrate of wealth on a given return process. In its most general form this port-folio strategy is allowed to adapt to the endogenous fluctuations of the re-turns. It then maximizes the expected logarithm of the returns, which isalso known as the Kelly rule, Kelly (1956). In this most general from it isclearly unbeatable in view of the long run perspective taken here. However,in this general form it is quite difficult to actually compute this rule. Oneway of interpreting the results of this paper is to say that with endogenousreturns there is a simple short-cut to determine a simple portfolio rule thateventually coincides with the Kelly rule: Usingλ∗ and thus simply dividingwealth proportional to the expected relative dividends. That is to say (cf.Corollary 1)

λgop(λ∗) = λ∗.

The alternative growth optimal strategy is then the one based on equalprices

λgop(1).

Appendix B collects the portfolio rules that have been computed accord-ing to the various strategies outlined so far. One apparent observation is thatthe portfolio strategies based on endogenous returns are under-diversified.The mean-variance-strategy only uses 8, the CVaR-strategy only uses 6and the growth optimal portfolio only uses 1 out of the 21 assets! As Hensand Schenk-Hoppé (2004) have shown in the case of short lived assets,under-diversification is fatal for survival in the market selection process,see Corollary 2. Therefore, we do the mean-variance rule yet another fa-vor and make it completely diversified by devoting to any asset at least the


smallest positive budget share occurring in the under-diversified portfolio6.This ad hoc diversification rule is often used in praxis:

λµ−σε (λ∗)

Thus all together we consider the market selection process given byequation (5) when it is run by these 8 portfolio rules.

5. Simulation Results

5.1 Rational Strategies can be driven out by Irrational Strategies

Friedman (1953) and Fama (1965) argued that the market naturally se-lects for the rational strategies. On the other hand De Long, Shleifer, Sum-mers, and Waldmann (1990) showed that under specific circumstancesnoise traders can earn a higher average rate of return than rational arbi-trageurs. In this section we demonstrate that at a first approximation theclaim of De Long, Shleifer, Summers and Waldmann (1990) can be givengood support in our model. We consider the fix-mix mean-variance port-folio rule λµ−σ(1) in competition with the illusionary diversification ruleλillu

k = (1−λ0)/K, k = 1, ..,K.

Figure 1 shows a typical run of the evolution of market shares over timefor a sample path of the dividend process. Starting with an initial distri-bution of wealth in which90%of the wealth is in the hands of the mean-variance rule, after only 10 periods the illusionary portfolio rule has con-quered more than90%of the market wealth. Thus the illusionary diversifi-cation rule quickly drives out the mean-variance rule. Moreover, after only15 periods the illusionary portfolio rule will hold almost all market wealth.Note that we have even given the mean-variance rule rational expectationsbecause we allowed it to be based on the prices that will eventually pre-vail in the market selection process. An intuition for this result is that themean-variance rule is under-diversified while the illusionary diversificationrule is completely diversified. Evstigneev, Hens and Schenk-Hoppé (2003)

6 Note that our notion of being completely diversified does not coincide with anintuitive notion of complete diversification, like naive diversification, according towhich your portfolio has equal shares in every strategy.


have shown that under-diversified rules cannot be evolutionary stable, butthat they can perform as badly as in this example is still surprising.

Figure 1: An irrational rule driving out a seemingly rational rule.Evolution of market shares: illusionary diversification rule (broken line)

and mean-variance rule forλillu -prices (bold line).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Market share λillu

Market share λµ−σ(λillu)

5.2 Stochastic Time Series of Asset Prices

Our next example shows that the market selection process studied here canalso generate stochastic time series of asset prices that do not converge.While the asset price dynamics in the standard evolutionary asset pricingmodels is complex and therefore not straightforward to interpret, the ex-planation of excess volatility arising in our model is quite simple. In ourmodel it is the endogenous change in the wealth shares that generates thefluctuations in asset prices. In an example with two strategies – the mean-conditional-value at risk strategyλCVaR(λ∗) and the illusionary diversifica-tion ruleλillu – the following phenomenon occurs. As the wealth share ofthe first strategy increases, it turns market prices to its disadvantage giv-ing the second strategy a higher potential for growth. And the same holds


true when the second strategy’s wealth share increases. Note that in ourmodel relative asset pricesqk

t are the wealth average of the strategies in themarket:

qkt =

I

∑i=1

λikr

it .

Hence with time independent strategiesλi price fluctuations can only resultfrom wealth fluctuations.

Figure 2 shows the evolution of market shares over time for one samplepath of the dividend process. Starting with initial wealth at40% wealthin the hands of the mean-conditional-value at risk rule, the wealth processcycles irregularly between35%and48%of wealth for this rule and has notyet converged after 1000 periods.

Figure 2: Non-convergence of the market selection process. Evolution ofmarket shares: illusionary diversification rule (broken line) and CVaR

rule (bold line).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 100 200 300 400 500 600 700 800 900

Market share λillu

Market share λCV aR


5.3 Single Survivor Hypothesis

In this subsection we now includeλ∗ in the simulations. Given the dividendmatrix and the strategies described above, we have carried out simulationsof the market selection process with different initial wealth shares forλ∗

and different number of periods. It turns out thatλ∗ satisfies the singlesurvivor property first defined in Blume and Easley (1992). On almost allpaths the wealth ofλ∗ grows at a faster rate than the wealth of any otherstrategy that we considered.

Figure 3 shows the evolution of market shares over time for one samplepath of the dividend process. Starting with equal initial wealth, after 100periods the evolutionary portfolio ruleλ∗ has conquered 50% of the marketwealth. We can see from Figure 3 that the market share ofλ∗ increasessteadily from10%to 50%while the other strategies’ market shares have aclear downward trend although some of them initially increase. Note thatafter less than 10 periodsλ∗ has doubled its market share for the first timeand that after less than 60 periods it has doubled it again.

Figure 3: Evolution of market shares: Typical sample path.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90 100

λ∗

λcpt

λillu

λCV aR(λ∗)

@@

@@

@@

@@

@@

@@

λCV aR(1)

SS

SS

SS

SS

SSS

λgop(1)

AAAAAAAAAA

λµ−σ(1)

DDDDDDDDDD

λµ−σ(λ∗)

EEEEEEE

Figure 4 shows the mean market shares, averaged over 30 runs. Equalinitial wealth is given to each of the 10 strategies described above. Eachrun was conducted for 100 periods.


Numerical studies show that the standard deviation of each strategiesmarket share from the mean in anyone period is quite small with a maxi-mum value of3%and an average value of only0.36%.

Figure 4: Evolution of average market shares: Average taken over 30runs.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90 100

λ∗

λcpt

λillu

λCV aR(λ∗)

@@

@@

@@

@@

@@

@@

λCV aR(1)

SS

SS

SS

SS

SSS

λgop(1)

AAAAAAAAAA

λµ−σ(1)

DDDDDDDDDD

λµ−σ(λ∗)

EEEEEEE

Figure 5 reports the prices that are implied by the evolution of marketshares. It is astonishing to see that prices converge quite rapidly to their ra-tional values which are determined byλ∗. Exxon Mobil Corp. is the com-pany with the highest price as it pays out the highest relative dividend onaverage.

Figure 6 depicts the evolution of market shares whenλ∗ starts with acomparative disadvantage. Initially it has only0.1% of total wealth. Thisfigure displays an interesting population dynamics. As long asλ∗ is small,its behavioral finance variationλcpt drives out the other strategies. How-ever,λ∗ grows steadily and eventually drives out and replacesλcpt. Notethat the chart ofλ∗ is S-shaped. Whileλ∗ is small it grows slowly, thenit has a rapid take off and eventually – when more and more competitorsget close to extinction – it slows down again. Even thoughλ∗ needs sometime to conquer a considerable share of the market, starting from the0.1%level it is able to double its share more rapidly than starting from the10%


Figure 5: Convergence of relative stock prices as time elapses for one run.Each sample path is the time series of one firm’s stock price.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50 60 70 80 90 100

Exxon Mobil Corp.

level. After 4 periods is has doubled for the first time, after 17 periods ithas doubled for the second time and after 40 periods it has doubled for thethird time. All other strategies only play a minor role in this dynamics.

This subsection has demonstrated that a rational investor should choosethe strategyλ∗. He will then drive out any other strategy. Hence, eventhough some seemingly rational strategies may do worse than some ir-rational strategies, the true rational strategy will always do better than anyirrational strategy. In this sense Friedman (1953) and Fama (1965) are right– eventually asset prices are as in an efficient market. They are determinedonly by expected relative dividends.

6. Some Intuition for the Results

To provide some intuition for the striking results obtained in the previoussection, it is instructive to recall the results of the theoretical literature.This literature considers an investment problem with stationary dividendsin which the returns at any point in time are completely re-invested for thenext period. The starting point was Breiman (1961)’s observation that the


Figure 6: Evolution of market shares: Initial market share ofλ∗ is 0.1%.Initially the behavioral finance ruleλcpt performs best.λ∗’s market sharegrows steadily. Eventuallyλ∗ drives out and replacesλcpt. All other rules

play a minor role.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700 800 900 1000

λ∗

λcpt

λillu

λCV aR(λ∗)

best strategy for repeatedly betting on the occurrence of a finite number ofstates is to divide the wealth placed on these bets proportional to the prob-abilities of occurrence of the states. This rule has thus been calledbettingyour beliefs. That is to say, if one holds fixed these proportions then, bythe Law of Large Numbers, you will maximize the expected growth rateof your wealth. Note that taking the long run perspective, risk does notmatter because any short run under-performance can still be recovered inthe long run. This point of view on the risk involved in portfolio formationis common to all papers on evolutionary portfolio theory. The next step inthis literature was to consider a market for the bets on the various states.Thus if demand for any one bet were high then the price for this bet will behigh and one might argue that one should rather go for the other bets thatoffer a more attractive return. However, as Blume and Easley (1992) haveshown, this is not true. The best portfolio rule is still to bet your beliefs. InBreiman (1961) (as well as in Blume and Easley (1992)) bets can be iden-tified with states because they consider a complete set of Arrow-securities.Evstigneev, Hens and Schenk-Hoppé (2002) have generalized the set up


of Blume and Easley (1992) to allow for any complete or incomplete as-set structure. As these authors show, the correct generalization of bettingyour beliefs is then to divide income proportionally to the expected pay-offs of the securities. A major shortcoming of the literature so far was theassumption of short-lived assets. According to this assumption the asset isliquidated after having paid off and an identical asset is born. Wealth isassumed to be perishable so that it can only be transferred to later peri-ods by investing once more in the exogenously supplied assets. Sandroni(2000) and Blume and Easley (2001) present the first models of this litera-ture with long-lived assets allowing the important feature of capital gains.However, these authors assume a complete security market and moreoverthey restrict attention to portfolio rules being generated by expected utilitymaximizers. This paper has a model with long-lived assets and a generalsecurity market. Moreover, portfolio rules need not be generated by ex-pected utility maximization. As it turns out, the wealth process convergesto the evolutionary portfolio ruleλ∗ and therefore capital gains convergeto the dividend payoffs. Hence the strategy being best suited to the div-idends will eventually also profit most from the capital gains. Note thatonly the evolutionary ruleλ∗ found in Hens and Schenk-Hoppé (2004) hasthis property so that in the long run this strategy has the highest expectedgrowth rate.

Let us finally compare the evolutionary portfolio ruleλ∗ with the CAPMrule, λCAPM. According to the CAPM, a passive buy and hold strategy,one should hold a fixed fraction of the market portfolio. In the notationof this paper, this would translate to having the demandaCAPM

t,k = γt , where

γt = (∑k pkt )−1 is some positive scalar. In terms of budget shares the CAPM

strategy is given byλCAPMt,k = γt pk

t , k = 1, ...K.The first observation is that in a rational and risk neutral market,λ∗

would actually coincide with the CAPM rule because in such a market assetprices are determined by discounted expected dividends, i.e.pk = 1

r fEdk,

k = 1, ...,K, wherer f denotes the risk free rate of interest. Asλ∗ gains totalmarket wealth, prices converge to the rational and risk neutral valuationand thusλ∗ and the CAPM rule will eventually coincide. Hence whileλ∗

exploits the wealth of other strategies it will never be able to drive out theCAPM rule. In a sense, the CAPM rule is a imitation strategy that mimicsthe best performing strategy in the long run.


It is noteworthy that, similar to contrarian strategies from behavioralfinance, the evolutionary portfolio rule eventually eliminates the marketanomaly from which it lives. As long aspk 6= 1

r fEdk,k = 1, ...,K there are

excess returns and henceλ∗ can grow at the expense of the existing ones.In the limit, as the distribution of wealth concentrates onλ∗, these excessreturns are removed.

7. Conclusions

Our simulations have shown that in competition with fix-mix rules derivedfrom mean-variance-optimization, from maximum growth theory and frombehavioral finance, the evolutionary portfolio rule discovered in Hens andSchenk-Hoppé (2004) will eventually hold total market wealth. Accord-ing to this simple rule the portfolio weights should be proportional to theexpected relative dividends of the assets. This rule may be interpreted asa CAPM rule which fixes budget shares according to the expected mar-ket capitalization and then rebalances according to these fixed weights asprices fluctuate. For sufficiently patient investors, like pension funds or in-surance companies for example, this rule promises very high proceeds. Ona long horizon risk is limited – the standard deviations of the average po-sition of the market share are very small and almost constant over time.Risk will however matter if the investor is subject to shocks that require toliquidate some of its wealth at unforeseen periods. An important and alsovery interesting extension of this work is to introduce such liquidity shocksin the evolutionary process of market selection.


A. The Dividend Process

Figure 7 depicts the evolution of relative dividends-per-share over time.Some firms apply dividend smoothing and distribute an almost constantstream of dividends while other firms’ dividend payments vary consider-ably. The dividends are adjusted for buy backs. In the simulations we haveidentified each year with a state of the world and then we have drawn suchstates independently and identically distributed according to a uniform dis-tribution.

Figure 7: The relative dividend process for the DJIA (21 states).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

AT&T roughly pays about one third of the total dividends in 1981-1984(states 1-4). Exxon Mobil Corp. has a roughly constant share of 15-20% ofthe total dividend payments over the entire period.

Company Name (Ticker Symbol): ALCOA Inc. (AA), American Ex-press Co. (AXP), AT&T Corp. (T), Boeing Co. (BA), Caterpillar Inc.(CAT), Coca-Cola Co. (KO), Dupont Co. (DD), Eastman Kodak Co. (EK),Exxon Mobil Corp. (XOM), General Electric Co. (GE), General MotorsCorp. (GM), Hewlett Packard Co. (HPQ), International Business MachinesCorp. (IBM), International Paper Co. (IP), J.P. Morgan Chase & Co. (JPM),McDonalds Corp. (MCD), Merck & Co. (MRK), Minnesota Mining &Manufacturing Co. (MMM), Phillip Morris Co. (MO), Procter & GambleCo. (PG), United Technologies Corp. (UTX).


AA AXP T BA CAT KO DD1981 0.1168 0.1440 3.7699 0.1349 0.2010 0.2669 0.41301982 0.1304 0.1777 4.2293 0.1298 0.2011 0.2755 0.48901983 0.1152 0.1905 4.2011 0.1196 0.1869 0.2848 0.51021984 0.0783 0.2014 4.4768 0.1083 0.1099 0.2905 0.48171985 0.0741 0.2081 1.7295 0.1011 0.0892 0.2706 0.52401986 0.0703 0.2166 0.9664 0.1104 0.0345 0.2735 0.51401987 0.0695 0.2100 0.9118 0.1228 0.0324 0.2658 0.49121988 0.0660 0.2072 0.8091 0.1330 0.0306 0.4638 0.49161989 0.0669 0.1917 0.7359 0.1352 0.0439 0.2528 0.51061990 0.1311 0.2013 0.6961 0.1451 0.0653 0.2647 0.55421991 0.1397 0.2275 0.7334 0.1716 0.0638 0.2891 0.57851992 0.0760 0.2364 0.7745 0.1700 0.0600 0.3172 0.56341993 0.0655 0.2442 0.8240 0.1603 0.0283 0.3479 0.55721994 0.0632 0.2335 0.7875 0.1509 0.0271 0.3920 0.53311995 0.0611 0.2133 0.7912 0.1439 0.0385 0.4257 0.52761996 0.0651 0.1836 0.8368 0.1371 0.0958 0.4449 0.47931997 0.0883 0.1645 0.8005 0.1430 0.1090 0.4704 0.47571998 0.0609 0.1512 0.7657 0.1991 0.1208 0.4958 0.50081999 0.0897 0.1401 0.7399 0.1908 0.1353 0.5007 0.52402000 0.0944 0.1280 0.9398 0.1701 0.1410 0.5006 0.47882001 0.1276 0.1285 1.0200 0.1539 0.1411 0.5144 0.4473

EK XOM GE GM HPQ IBM IP1981 0.5164 2.3484 0.6700 0.8741 0.0240 2.0080 0.13981982 0.5438 2.4925 0.6869 0.7018 0.0259 1.9436 0.13951983 0.5147 2.3071 0.6732 0.6645 0.0266 1.8186 0.12851984 0.4674 2.1290 0.6784 0.7104 0.0318 1.7923 0.11561985 0.4296 2.0372 0.6912 1.1325 0.0364 1.8633 0.10851986 0.3889 1.8333 0.7173 1.1370 0.0401 1.9008 0.10201987 0.3638 1.7187 0.6986 1.0981 0.0370 1.7814 0.09181988 0.3506 1.6465 0.7215 1.0224 0.0368 1.6269 0.09071989 0.3423 1.6378 0.7205 0.9455 0.0394 1.4883 0.09361990 0.3502 1.5874 0.7980 1.0598 0.0459 1.4849 0.10091991 0.3395 1.6514 0.8777 1.0234 0.0534 1.4510 0.09571992 0.3216 1.6863 0.8821 0.5760 0.0595 1.3732 0.09221993 0.3064 1.6853 0.9075 0.6490 0.0863 1.3035 0.09711994 0.2916 1.6113 0.9557 0.4811 0.1012 0.4141 0.09231995 0.2395 1.5482 1.0417 0.4705 0.1185 0.2801 0.08891996 0.2192 1.5089 1.1101 0.5321 0.1435 0.2369 0.09501997 0.2033 1.4719 1.1505 0.5772 0.1698 0.2663 0.10981998 0.2027 1.4434 1.2193 0.5791 0.1902 0.2799 0.10801999 0.1925 1.3572 1.3238 0.4696 0.2114 0.2821 0.10352000 0.1784 1.8605 1.4534 0.4331 0.2059 0.2785 0.13242001 0.1664 1.8694 1.6490 0.3951 0.1948 0.2836 0.1365


JPM MCD MRK MMM MO PG UTX1981 0.0595 0.0297 0.1781 0.3288 0.1995 0.2811 0.17571982 0.0695 0.0368 0.1891 0.3382 0.2402 0.3017 0.18731983 0.0919 0.0419 0.1835 0.3331 0.2671 0.3012 0.17441984 0.1033 0.0462 0.1678 0.3089 0.2913 0.2970 0.16571985 0.1115 0.0520 0.1665 0.2958 0.3093 0.2966 0.16331986 0.1095 0.0565 0.1653 0.2834 0.3368 0.3059 0.15831987 0.1029 0.0568 0.1839 0.2720 0.3896 0.2938 0.13881988 0.1300 0.0574 0.2053 0.2605 0.4591 0.2856 0.11251989 0.1369 0.0585 0.2879 0.2744 0.5106 0.2721 0.11551990 0.1484 0.0621 0.3509 0.3119 0.5941 0.2806 0.12271991 0.1778 0.0697 0.3920 0.3384 0.7067 0.3343 0.13261992 0.2007 0.0735 0.4426 0.3394 0.8315 0.3731 0.13921993 0.2065 0.0757 0.5017 0.3305 0.9560 0.3715 0.12451994 0.2131 0.0893 0.5213 0.3200 1.0169 0.3773 0.11851995 0.2204 0.0913 0.6068 0.3148 1.0523 0.4015 0.10071996 0.3920 0.0908 0.6171 0.3166 1.1779 0.4256 0.10101997 0.4481 0.0875 0.6522 0.3029 1.3060 0.4534 0.10001998 0.4332 0.0885 0.7292 0.3131 1.3887 0.4751 0.10401999 0.4323 0.0814 0.7622 0.3001 1.3478 0.4946 0.10692000 0.4442 0.0839 0.8205 0.2855 1.3745 0.5152 0.11182001 0.6967 0.0857 0.8542 0.2803 1.3739 0.5483 0.1182


B. Portfolio Rules

The following table reports the budget shares for the investment strategiesapplied in this paper. The budget shares are normalized with(1−λ0) forconvenience. Rounding errors may prevent shares from adding up to one.

Weights (%) λµ−σ(λ∗) λCVar(λ∗) λ∗ λgop(1) λillu λcpt(λ∗)UTX 0.00 0.00 1.28 0.00 4.76 2.82PG 0.00 0.00 3.63 0.00 4.76 4.70MO 0.00 27.45 7.81 0.00 4.76 6.58

MMM 0.00 0.00 3.02 0.00 4.76 4.08MRK 0.46 0.00 4.31 0.00 4.76 5.64MCD 91.01 0.00 0.68 0.00 4.76 2.19JPM 0.00 0.00 2.35 0.00 4.76 3.45IP 0.00 0.00 1.04 0.00 4.76 2.51

IBM 0.07 18.56 10.97 0.00 4.76 7.52HPQ 3.79 0.00 0.90 0.00 4.76 2.51GM 0.22 18.22 7.31 0.00 4.76 6.58GE 0.10 0.00 9.28 0.00 4.76 7.21

XOM 0.00 17.26 17.25 99.99 4.76 8.46EK 3.44 0.00 3.17 0.00 4.76 5.02DD 0.00 4.26 5.02 0.00 4.76 5.96KO 0.91 9.71 3.61 0.00 4.76 5.02CAT 0.00 0.00 0.86 0.00 4.76 2.51BA 0.00 0.00 1.43 0.00 4.76 3.13T 0.00 4.54 13.33 0.01 4.76 8.15

AXP 0.00 0.00 1.90 0.00 4.76 3.45AA 0.00 0.00 0.85 0.00 4.76 2.51

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Q UANTITATIVE F I N A N C E V O L U M E 1 (2001) 149–167 RE S E A R C H PA P E RI N S T I T U T E O F P H Y S I C S P U B L I S H I N G quant.iop.org

Financial markets as nonlinearadaptive evolutionary systems

Cars H Hommes

Center for Nonlinear Dynamics in Economics and Finance (CeNDEF)1,Department of Economics, University of Amsterdam, Roetersstraat 11,NL-1018 WB Amsterdam, The Netherlands

E-mail: [email protected]

Received 15 October 2000

AbstractRecent work on complex adaptive systems for modelling financial markets issurveyed. Financial markets are viewed as evolutionary systems betweendifferent, competing trading strategies. Agents are boundedly rational in thesense that they tend to follow strategies that have performed well, accordingto realized profits or accumulated wealth, in the recent past. Simple technicaltrading rules may survive evolutionary competition in a heterogeneous worldwhere prices and beliefs coevolve over time. The evolutionary modelexplains stylized facts, such as fat tails, volatility clustering and long memory,of real financial series. Although our adaptive belief systems are very simple,they can match the autocorrelation patterns of returns, squared returns andabsolute returns of 40 years of S&P 500 data. Some recent laboratory workon expectation formation in an asset pricing framework is also discussed.

1. IntroductionThe key difference between economics and the natural sciencesis perhaps the fact that decisions of economic agents todaydepend upon their expectations or beliefs about the future.For example, after a couple of weeks of bad weather in theNetherlands and Western Europe in July 2000, the dreams andhopes of the Dutch about nice weather for summer holidayswill not affect the weather in August. In contrast, the dreamsand hopes of Dutch investors for excessive high returns on theirinvestments in tulip bulbs in the seventeenth century may havecontributed to or even caused what is nowadays known as theDutch ‘tulip mania’, when the price of tulip bulbs exploded bya factor of more than 20 in the beginning of 1636 but ‘crashed’back to its original level by the end of the year. Nowadays, infinancial markets an over-optimistic estimate of future growthof ICT industries may contribute to an excessively rapid growthof stock prices and indices and might lead to over valuation ofstock markets worldwide. Any dynamic economic system is infact an expectations feedback system. A theory of expectation

1 Web address: http://www.fee.uva.nl/cendef

formation is therefore a crucial part of any economic model ortheory.

Since its introduction in the sixties by Muth (1961)and its popularization in macroeconomics by Lucas (1971),the rational expectations hypothesis (REH) has become thedominating expectation formation paradigm in economictheory. According to the REH all agents are rational andtake as their subjective expectation of future variables theobjective prediction by economic theory. In a rationalexpectations model agents have perfect knowledge about the(linear) market equilibrium equations and use these to derivetheir expectations. Although many economists nowadaysview rational expectations as something unrealistic, it is stillviewed as an important benchmark. Despite a rapidly growingliterature on bounded rationality, where agents use learningmodels for their expectations, it seems fair to say that at thispoint no generally accepted alternative theory of expectationsis available.

In finance, the REH is intimately related to the efficientmarket hypothesis (EMH). There are weak and strong formsof the EMH, but when economists speak of financial markets

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C H Hommes QUANTITATIVE FI N A N C E

as being efficient, they usually mean that they view assetprices and returns as the outcome of a competitive marketconsisting of rational traders, who are trying to maximizetheir expected returns. The main reason why financial marketsmust be efficient is based upon an arbitrage argument (e.g.Fama (1970)). If markets were not efficient, then there wouldbe unexploited profit opportunities, that could and would beexploited by rational traders. For example, rational traderswould buy (sell) an underpriced (overpriced) asset, thus drivingits price back to the correct, fundamental value. In anefficient market, there can be no forecastable structure inasset returns, since any such structure would be exploited byrational traders and therefore would be doomed to disappear.Rational agents thus process information quickly and this isreflected immediately in asset prices. The value of a riskyasset is completely determined by its fundamental price, equalto the present discounted value of the expected stream of futuredividends. In an efficient market, all traders are rational andchanges in asset prices are completely random, solely driven byunexpected ‘news’ about changes in economic fundamentals.

In contrast, Keynes (1936) already questioned acompletely rational valuation of assets, arguing that investorssentiment and mass psychology (‘animal spirits’) play asignificant role in financial markets. Keynes used his famousbeauty contest as a parable about financial markets. In orderto predict the winner of a beauty contest, objective beauty isnot all that important, but knowledge or prediction of others’perceptions of beauty is much more relevant. Keynes arguedthat the same may be true for the fundamental price of anasset: ‘Investment based on genuine long-term expectation isso difficult as to be scarcely practicable. He who attempts itmust surely lead much more laborious days and run greaterrisks than he who tries to guess better than the crowd how thecrowd will behave; and, given equal intelligence, he may makemore disastrous mistakes’ (Keynes (1936) p 157). In Keynesview, stock prices are thus not governed by an objective view of‘fundamentals’, but by ‘what average opinion expects averageopinion to be’.

New classical economists have viewed ‘market psychol-ogy’ and ‘investors sentiment’ as being irrational however,and therefore inconsistent with the REH. For example, Fried-man (1953) argued that irrational speculative traders would bedriven out of the market by rational traders, who would tradeagainst them by taking long opposite positions, thus drivingprices back to fundamentals. In an efficient market, ‘irrational’speculators would simply lose money and therefore fail to sur-vive evolutionary competition.

Financial markets as nonlinear evolutionaryadaptive systems

In a perfectly rational EMH world all traders are rational andit is common knowledge that all traders are rational. In realfinancial markets however, traders are different, especially withrespect to their expectations about future prices and dividends.A quick glance at the financial pages of newspapers is sufficientto observe that difference of opinions among financial analystsis the rule rather than the exception. In the last decade, a

rapidly increasing number of structural heterogeneous agentmodels have been introduced in the finance literature, seefor example Arthur et al (1997), Brock (1993, 1997), Brockand Hommes (1997a, b, 1998), Brock and LeBaron (1996),Chiarella (1992), Chiarella and He (2000), Dacorogna et al(1995), DeGrauwe et al (1993), De Long et al (1990), Farmer(1998), Farmer and Joshi (2000), Frankel and Froot (1988),Gaunersdorfer (2000), Gaunersdorfer and Hommes (2000),Kirman (1991), Kirman and Teyssiere (2000), Kurz (1997),LeBaron (2000), LeBaron et al (1999), Lux (1995), Lux andMarchesi (1999a, b), Wang (1994) and Zeeman (1974), aswell as many more references in these papers. Some authorseven talk about a heterogeneous market hypothesis, as a newalternative to the efficient market hypothesis. In all theseheterogeneous agent models different groups of traders, havingdifferent beliefs or expectations, coexist. Two typical tradertypes can be distinguished. The first are rational, ‘smartmoney’ traders or fundamentalists, believing that the price ofan asset is determined completely by economic fundamentals.The second typical trader type are ‘noise traders’, sometimescalled chartists or technical analysts, believing that asset pricesare not determined by fundamentals, but that they can bepredicted by simple technical trading rules based upon patternsin past prices, such as trends or cycles.

In a series of papers, Brock and Hommes (1997a, b,1998, 1999), henceforth BH, propose to model economicand financial markets as adaptive belief systems (ABS). Thepresent paper reviews the main features of ABS and discussesa recent extension by Gaunersdorfer and Hommes (2000) aswell as some recent experimental testing, jointly with mycolleagues Joep Sonnemans, Jan Tuinstra and Henk van deVelden (Hommes et al 2000b) at CeNDEF. An ABS is anevolutionary competition between trading strategies. Differentgroups of traders have different expectations about futureprices and future dividends. For example, one group mightbe fundamentalists, believing that asset prices return to theirfundamental equilibrium price, whereas another group mightbe chartists, extrapolating patterns in past prices. Traderschoose their trading strategy according to an evolutionary‘fitness measure’, such as accumulated past profits. Agentsare boundedly rational, in the sense that most traderschoose strategies with higher fitness. BH introduce thenotion of adaptive rational equilibrium dynamics (ARED), anendogenous coupling between market equilibrium dynamicsand evolutionary updating of beliefs. Current beliefs determinenew equilibrium prices, generating adapted beliefs which inturn lead to new equilibrium prices again, etc. In an ARED,equilibrium prices and beliefs coevolve over time.

Most of the heterogeneous agent literature is computation-ally oriented. An ABS may be seen as a tractable theoreticalframework for the computationally oriented ‘artificial stockmarket’ literature, such as the Santa Fe artificial stock marketof Arthur et al (1997) and LeBaron et al (1999). A convenientfeature of an ABS is that the model can be formulated in termsof deviations from a benchmark fundamental. In fact, the per-fectly rational EMH benchmark is nested within an ABS asa special case. An ABS may thus be used for experimental

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QUANTITATIVE FI N A N C E Financial markets as nonlinear adaptive evolutionary systems

and empirical testing whether deviations from a suitable REbenchmark are significant.

The heterogeneity of expectations among tradersintroduces an important nonlinearity into the market. Inan ABS there are also two important sources of noise:model approximation error and intrinsic uncertainty abouteconomic fundamentals. Asset price fluctuations in anABS are characterized by an irregular switching betweenphases of close-to-the-fundamental-price fluctuations, phasesof optimism where most agents follow an upward price trend,and phases of pessimism with small or large market crashes.Temporary speculative bubbles (rational animal spirits) canoccur, triggered by noise and amplified by evolutionary forces.An ABS is able to generate some of the important stylized factsin many financial series, such as unpredictable returns, fat tailsand volatility clustering.

In our discussion of ABS we will focus on the followingquestions central to the SFI workshop:

Q1 Can technical analysts or habitual rule-of-thumb tradingstrategies survive evolutionary competition againstrational or fundamental traders?

Q2 Is an evolutionary adaptive financial market withcompeting heterogeneous agents efficient?

Q3 Does heterogeneity in beliefs lead to excess volatility?

The paper is organized as follows. In section 2 we discussthe modelling philosophy emphasizing recent developmentsin nonlinear dynamics and their relevance to economics andfinance. Section 3 presents ABS in a general mean-varianceframework. In section 4 we present simple, but typicalexamples. Although the ABS are very simple, subsection4.4 presents an example where the autocorrelations of returns,squared returns and absolute returns closely resemble those of40 years of S&P 500 data. Section 5 briefly discusses somefirst experimental testing of ABS. Finally, section 6 sketches afuture perspective of the research program proposed here.

2. Philosophy of nonlinear dynamicsThe past 25 years have witnessed an explosion of interestin nonlinear dynamical systems, in mathematics as wellas in applied sciences. In particular, the fact that simpledeterministic nonlinear systems exhibit bifurcation routes tochaos and strange attractors, with ‘random looking’ dynamicalbehaviour, has received much attention. This section discussessome important features of nonlinear systems, emphasizingtheir relevance to economics and finance. Let us start by statingthe main goal of our research program, namely to explain themost important ‘stylized facts’ in financial series, such as:

S1 Asset prices are persistent and have, or are close to having,a unit root.

S2 Asset returns are fairly unpredictable, and typically havelittle or no autocorrelations.

S3 Asset returns have fat tails and exhibit volatility clusteringand long memory. Autocorrelations of squared returns andabsolute returns are significantly positive, even at highorder lags, and decay slowly possibly following a scalinglaw.

S4 Trading volume is persistent and there is positive crosscorrelation between volatility and volume.

In this paper we will be mainly concerned with stylizedfacts S2 and S32. The adaptive belief system introduced in thenext section will be a nonlinear stochastic system of the form

Xt+1 = F(Xt ; n1t , . . . , nHt ; λ; δt ; εt ), (1)

where F is a nonlinear mapping, Xt is a vector of prices (orlagged prices), njt is the fraction or weight of investors oftype h, 1 � h � H , λ is a vector of parameters and δt andεt are noise terms. In an ABS there are two types of noiseterms which are relevant for financial markets. The noiseterm εt is the model approximation error representing the factthat a model can only be an approximation of the real world.Approximation errors will also be present in a physical model,although the corresponding noise terms might be of smallermagnitude than in economics. In contrast to physical modelshowever, in economic and financial models one almost alwayshas to deal with intrinsic uncertainty represented here by thenoise term δt . In finance, for example, one typically dealswith investors’ uncertainty about economic fundamentals. Inthe ABS there will be uncertainty about future dividends andthe noise term δt represents unexpected random news aboutdividends. An important goal of our research program isto match the nonlinear stochastic model (1) to the statisticalproperties of the data, as closely as possible, and in particularto first match the most important stylized facts in the data3.

A special case of the nonlinear stochastic system (1) ariseswhen all noise terms are set to zero. We will refer to this systemas the (deterministic) skeleton denoted by4

Xt+1 = F(Xt ; n1t , . . . , nHt ; λ). (2)

In order to understand the properties of the general stochasticmodel (1) it is important to understand the properties ofthe deterministic skeleton. In particular, one would like toimpose as little structure on the noise process as possible, andrelate the stylized facts of the general stochastic model (1)directly to generic properties of the underlying deterministicskeleton. There are three important, generic features ofnonlinear deterministic systems which may play an importantrole in generating some of the stylized facts in finance and mayin particular cause volatility clustering:

F1 Chaos and strange attractors due to homoclinicbifurcations.

F2 Simultaneous coexistence of different attractors.F3 Local bifurcations of steady states.

2 We will mainly focus on stationary systems here, but the example insubsection 4.4 will be close to having a unit root and matching the stylized factS1. In future work, we also plan to investigate the relation between tradingvolume and volatility in stylized fact S4.3 Other approaches to modelling the stylized facts, especially volatilityclustering and fat tails, in finance include the (G)ARCH models initiatedby Engle (1982), and more recently the multifractal modelling approach ofMandelbrot (1997, 1999) and the ‘scaling-law approach’ in econophysics assurveyed in Mantegna and Stanley (2000). These approaches are statisticallyor time-series oriented however and, although important and useful, lackstructural economic modelling.4 This terminology is used e.g. by Tong (1990)

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0.12

0.07

0.02

–0.03

–0.08

0.8

0.4

0

–0.4

–0.8

–0.8 –0.4 0 0.4 0.8 –0.09 –0.03 0.03 0.09 0.15

Figure 1. Strange attractor (a) of the adaptive belief system of subsection 4.3 and enlargement of its fractal structure (b).

Figure 1 illustrates feature F1 and shows an example of astrange attractor in the ABS discussed in subsection 4.3.

The motion on the strange attractor is highly unpre-dictable, with asset prices jumping irregularly over the com-plicated fractal set. Although research in nonlinear dynam-ics has been stimulated much by computer simulations in thepast 25 years, the ‘roots of chaos’ date back to the mathe-matician Henri Poincare at the end of the previous century5.Poincare introduced the notion of homoclinic orbits, in hisinvestigations of the three-body problem. Nowadays it is amathematical fact that homoclinic orbits are a key feature ofchaotic systems, and so-called homoclinic bifurcations lead-ing to strange attractors seem to be the rule rather than theexception. Brock and Hommes (1997a, 1998) have shownthat evolutionary adaptive systems with heterogeneous agentsusing competing trading strategies is a natural nonlinear worldfull of homoclinic bifurcations and strange attractors.

Nonlinear dynamic models can generate a wide varietyof irregular patterns. In particular, nonlinear dynamic modelscan generate any given autocorrelation pattern6. A nonlinear,chaotic model, buffeted with dynamic noise, with almost noautocorrelations in returns but at the same time persistence insquared returns, with slowly decaying autocorrelations, maythus provide a structural explanation of the unpredictability of

5 Gleick (1987) presents a nice overview of the history of nonlinear dynamics.A good introduction to the mathematics of ‘chaos’ for non-specialists isRuelle (1991). A mathematical treatment with recent advances in homoclinicbifurcation theory is e.g. Palis and Takens (1993).6 To see that a higher-dimensional chaotic map can generate any desiredautocorrelation structure, consider the nonlinear difference equation (xt , yt ) =(a1xt−1 + . . . + aLxt−L + yt−1, 1 − 2y2

t−1). As is well known, thesecond coordinate yt follows a chaotic process with zero mean and zeroautocorrelations at all lags. Since yt is generated independently of past valuesof xt , the series xt and yt are uncorrelated. The first coordinate xt thus followsa linear AR(L) process driven by a chaotic series with zero autocorrelationsat all lags, and thus has the desired autocorrelation structure.

asset returns and volatility clustering in financial assets. In fact,the phenomenon of intermittency, as introduced by Pomeauand Manneville (1980), is well suited as a description of thephenomenon of volatility clustering. Intermittency meanschaotic asset price fluctuations characterized by phases ofalmost periodic fluctuations irregularly interrupted by suddenbursts of erratic fluctuations. In an ABS intermittency occurscharacterized by close to the RE fundamental steady statefluctuations suddenly interrupted by price deviations from thefundamental triggered by technical trading.

The second generic feature F2, coexistence of attractors,is also naturally suited to describe volatility clustering. Inparticular, the ABS exhibits coexistence of a stable steady stateand a stable limit cycle. When buffeted with dynamic noise,irregular switching occurs between close to the fundamentalsteady state fluctuations, when the market is dominated byfundamentalists, and periodic fluctuations when the marketis dominated by chartists. It is important to note thatboth intermittency and coexistence of attractors are persistentphenomena, which are by no means special for our ABS, butoccur naturally in nonlinear dynamic models, and moreoverare robust with respect to and sometimes even reinforced bydynamic noise.

The third generic feature F3, a local bifurcation of a steadystate, that is, a change in the stability of the steady state, isrelated to feature F2. A local bifurcation occurs when thelinearized system is at the border of stability, having at least oneunit root. Close to the bifurcation point there can be regions inthe parameter space where the nonlinear system has coexistingattractors. It turns out that an ABS with parameters close toa local bifurcation point of the steady state can generate someof the stylized facts in finance such as volatility clustering.

In the last decade several attempts have been made totest for chaos in economic and financial data. Most empirical

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studies have rejected the hypothesis that economic or financialdata are generated by low-dimensional, purely deterministicchaos but strong evidence for nonlinearity is found (see e.g. thesurvey in Brock et al (1991)). It should be stressed though thatthe methods for detecting chaos are very sensitive to noise, andthat nonlinear models with noise, such as our proposed ABS,may be consistent with the data. A noisy nonlinear model mayexplain a significant part of observed fluctuations and stylizedfacts in economic and financial markets.

3. Adaptive belief systemsThis section reviews the notion of an adaptive belief system(ABS), as introduced in Brock (1997) and Brock andHommes (1997a, b, 1998). An ABS is in fact a standarddiscounted value asset pricing model derived from mean-variance maximization, extended to the case of heterogeneousbeliefs. Agents can either invest in a risk free asset or in a riskyasset. The risk free asset is perfectly elastically supplied andpays a fixed rate of return r; the risky asset, for example a largestock or a market index, pays an uncertain dividend. Let pt bethe price per share (ex-dividend) of the risky asset at time t ,and let yt be the stochastic dividend process of the risky asset.Wealth dynamics is given by

Wt+1 = (1 + r)Wt + (pt+1 + yt+1 − (1 + r)pt )zt , (3)

where bold face variables denote random variables at datet + 1 and zt denotes the number of shares of the risky assetpurchased at date t . Let Et and Vt denote the conditionalexpectation and conditional variance based on a publicallyavailable information set such as past prices and past dividends.Let Eht and Vht denote the ‘beliefs’ or forecasts of tradertype h about conditional expectation and conditional variance.Agents are assumed to be myopic mean-variance maximizersso that the demand zht of type h for the risky asset solves

Maxzt{Eht [Wt+1] − a

2Vht [Wt+1]

}, (4)

where a is the risk aversion parameter. The demand zht forrisky assets by trader type h is then

zht = Eht [pt+1 + yt+1 − (1 + r)pt ]

aVht [pt+1 + yt+1 − (1 + r)pt ]

= Eht [pt+1 + yt+1 − (1 + r)pt ]

aσ 2, (5)

where the conditional variance Vht = σ 2 is assumed to beequal and constant for all types7. Let zs denote the supply ofoutside risky shares per investor, assumed to be constant, andlet nht denote the fraction of type h at date t . Equilibrium ofdemand and supply yields

H∑h=1

nhtEht [pt+1 + yt+1 − (1 + r)pt ]

aσ 2= zs, (6)

7 Gaunersdorfer (2000) investigates the case with time varying beliefs aboutvariances and shows that the results are quite similar to those for constantvariance.

where H is the number of different trader types. BH focus onthe special case of zero supply of outside shares, i.e. zs = 0, forwhich the market equilibrium equation can be rewritten as8,9

(1 + r)pt =H∑h=1

nhtEht [pt+1 + yt+1]. (7)

3.1. The EMH benchmark with rational agents

Let us first discuss the EMH benchmark with rationalexpectations. In a world where all traders are identicaland expectations are homogeneous the arbitrage marketequilibrium equation (7) reduces to

(1 + r)pt = Et [pt+1 + yt+1], (8)

where Et denotes the common conditional expectation of alltraders at the beginning of period t , based on a publicallyavailable information set It such as past prices and dividends,i.e. It = {pt−1, pt−2, . . . ; yt−1, yt−2, . . .}. This arbitragemarket equilibrium equation (8) states that today’s price ofthe risky asset must be equal to the sum of tomorrow’sexpected price and expected dividend, discounted by the risk-free interest rate. It is well known that, using the arbitrageequation (8) repeatedly and assuming that the transversalitycondition

limk→∞

Et [pt+k](1 + r)k

= 0 (9)

holds, the price of the risky asset is uniquely determined by

p∗t =

∞∑k=1

Et [yt+k](1 + r)k

. (10)

The price p∗t in (10) is called the EMH fundamental rational

expectations (RE) price, or the fundamental price for short.The fundamental price is completely determined by economicfundamentals and given by the discounted sum of expectedfuture dividends. In general, the properties of the fundamentalprice p∗

t depend upon the stochastic dividend process yt . Wewill mainly focus on the case of an IID dividend process yt ,with constant mean E[yt ] = y. We note however that anyother random dividend process yt may be substituted in whatfollows10. For an IID dividend process yt with constant mean,the fundamental price is constant and given by

p∗ =∞∑k=1

y

(1 + r)k= y

r. (11)

8 Brock (1997) motivates this special case by introducing a risk adjusteddividend y#

t+1 = yt+1 − aσ 2zs to obtain the market equilibrium equation(7). In general however, the equilibrium equation (7) ignores a risk premiumaσ 2zs for investors holding the risky asset. Since dividends and a risk premiumaffect realized profits and wealth, in general they will affect the fractions nht oftrader type h. The question how exactly the risk premium affects evolutionarycompetition will be investigated in future work, by taking zs as a bifurcationparameter.9 In the examples of ABS in section 4, we will add a noise term εt to theright-hand side of the market equilibrium equation (7), representing a modelapproximation error.10 Brock and Hommes (1997b) for example discuss a non-stationary example,where the dividend process is a geometric random walk.

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There are two crucial assumptions underlying the derivationof the RE fundamental price. The first is that expectationsare homogeneous, all traders are rational and it is commonknowledge that all traders are rational. In such an ideal,perfectly rational world the fundamental price can be derivedfrom economic fundamentals. Conditions under which aRE price can be derived can be relaxed, to include forexample noise traders or limited heterogeneity of information.In general however, in a world with heterogeneous tradershaving different beliefs or expectations about future pricesand dividends, derivation of a RE fundamental price requiresperfect knowledge about the beliefs of all other traders11. Ina real market, understanding the beliefs and strategies of allother, competing traders is virtually impossible, and thereforein a heterogeneous world derivation of the RE-fundamentalprice becomes impossible. The second crucial assumptionunderlying the derivation of the fundamental price is thetransversality condition (9), requiring that the long-run growthrate of prices (and risk adjusted dividends) is smaller than therisk free growth rate r . In fact, in addition to the fundamentalsolution (10) so-called speculative bubble solutions of the form

pt = p∗t + (1 + r)t (p0 − p∗

0) (12)

also satisfy the arbitrage equation (8). It is important to notethat along the speculative bubble solution (12), traders haverational expectations. Solutions of the form (12) are thereforecalled rational bubbles. These rational bubble solutions areexplosive and do not satisfy the transversality condition. Ina perfectly rational world, traders realize that speculativebubbles cannot last forever and therefore they will neverget started and the finite fundamental price p∗

t is uniquelydetermined. In a perfectly rational world, all traders thusbelieve that the value of a risky asset equals its fundamentalprice forever. Changes in asset prices are solely driven byunexpected changes in dividends and random ‘news’ abouteconomic fundamentals. In a heterogeneous evolutionaryworld however, the situation will be quite different, and we willsee that evolutionary forces may lead to endogenous switchingbetween the fundamental price and the rational self fulfillingbubble solutions.

3.2. Heterogeneous beliefs

In the asset pricing model with heterogeneous beliefs, marketequilibrium in (7) states that the price pt of the risky assetequals the discounted value of tomorrow’s expected price plustomorrow’s expected dividend, averaged over all differenttrader types. In such a heterogeneous world temporaryupward or downward bubbles with prices deviating from thefundamental may arise, when the fraction of traders believingin those bubbles is large enough. Once a (temporary) bubblehas started, evolutionary forces may reinforce deviations fromthe benchmark fundamental. We shall now be more preciseabout traders’ expectations (forecasts) about future prices anddividends. It will be convenient to work with

xt = pt − p∗t , (13)

11 See e.g. Arthur (1995) for a lucid account of this point.

the deviation from the fundamental price. We make thefollowing assumptions about the beliefs of trader type h:

B1 Vht [pt+1 + yt+1 − (1 + r)pt ] = Vt [pt+1 + yt+1 − (1 +r)pt ] = σ 2, for all h, t .

B2 Eht [yt+1] = Et [yt+1], for all h, t .B3 All beliefs Eht [pt+1] are of the form

Eht [pt+1] = Et [p∗t+1] + fh(xt−1, . . . , xt−L), for all h, t.

(14)

According to assumption B1, beliefs about conditionalvariance are equal and constant for all types, as discussedabove already. Assumption B2 states that expectations aboutfuture dividends yt+1 are the same for all trader types and equalto the conditional expectation. All traders are thus able toderive the fundamental price p∗

t in (10) that would prevail in aperfectly rational world. According to assumption B3, tradersnevertheless believe that in a heterogeneous world prices maydeviate from their fundamental value p∗

t by some functionfh depending upon past deviations from the fundamental.Each forecasting rule fh represents the model of the marketaccording to which type h believes that prices will deviatefrom the commonly shared fundamental price. For example, aforecasting strategy fh may correspond to a technical tradingrule, based upon short run or long run moving averages, of thetype used in real markets.

Strictly speaking (14) is not a technical trading rule,because it uses the fundamental price in its forecast. Includingprice forecasting rules depending upon past prices only, notusing any information about fundamentals, yields similarresults. However, as will be seen below, a convenient featureof our formulation is that the complete heterogeneous agentasset pricing model can be reformulated in terms of deviationsfrom the benchmark fundamental. We will use the short-handnotation

fht = fh(xt−1, . . . , xt−L) (15)

for the forecasting strategy employed by trader type h.Brock and Hommes (1998) have investigated evolutionarycompetition between the simplest linear trading rules with onlyone lag, i.e.

fht = ghxt−1 + bh. (16)

Simple forecasting rules are more likely to be relevant in realmarkets, because for a complicated forecasting rule it seemsunlikely that enough traders will coordinate on that particularrule so that it affects market equilibrium prices. Although thelinear forecasting rule (16) is extremely simple, it does in factrepresent a number of important cases. For example, whenboth the trend parameter and the bias parameter gh = bh = 0the rule reduces to the forecast of fundamentalists, i.e.

fht ≡ 0, (17)

believing that the market price will be equal to the fundamentalprice p∗, or equivalently that the deviation x from thefundamental will be 0. Other important cases covered by thelinear forecasting rule (16) are the pure trend followers

fht = ghxt−1, gh > 0, (18)

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and the pure biased belief

fht = bh. (19)

Notice that the simple pure bias (19) rule represents anypositively or negatively biased price forecast that traders mighthave. Instead of these extremely simple habitual rule of thumbforecasting rules, some economists might prefer the rational,perfect foresight forecasting rule

fht = xt+1. (20)

We emphasize however, that the perfect foresightforecasting rule (20) assumes perfect knowledge of theheterogeneous market equilibrium equation (7), and inparticular perfect knowledge about the beliefs of all othertraders. Although the case with perfect foresight certainlyhas theoretical appeal, its practical relevance in a complexheterogeneous world should not be overstated since thisunderlying assumption seems highly unrealistic12.

An important and convenient consequence of theassumptions B1–B3 concerning traders’ beliefs is that theheterogeneous agent market equilibrium equation (7) can bereformulated in deviations from the benchmark fundamental.In particular, substituting the price forecast (14) in themarket equilibrium equation (7) and using the facts that thefundamental price p∗

t satisfies (1 + r)p∗t = Et [p∗

t+1 + yt+1]and the price pt = xt + p∗

t yields the equilibrium equation indeviations from the fundamental:

(1 + r)xt =H∑h=1

nhtEht [xt+1] ≡H∑h=1

nhtfht , (21)

with fht = fh(xt−1, . . . , xt−L). An important reason for ourmodel formulation in terms of deviations from a benchmarkfundamental is that in this general setup, the benchmarkrational expectations asset pricing model is nested as a specialcase, with all forecasting strategies fh ≡ 0. In this way,the adaptive belief systems can be used in empirical andexperimental testing whether asset prices deviate significantlyfrom anyone’s favourite benchmark fundamental.

3.3. Evolutionary dynamics

The evolutionary part of the model describes how beliefs areupdated over time, that is, how the fractions nht of tradertypes in the market equilibrium equation (21) evolve over time.Fractions are updated according to an evolutionary fitness orperformance measure. The fitness measures of all tradingstrategies are publically available, but subject to noise. Fitnessis derived from a random utility model and given by

Uht = Uht + εht , (22)

where Uht is the deterministic part of the fitness measure andεht represents noise. Assuming that the noise εht is IID acrossh = 1, . . . , H drawn from a double exponential distribution, in

12 See also subsection 4.1 for a brief discussion of rational versusfundamentalist traders.

the limit as the number of agents goes to infinity, the probabilitythat an agent chooses strategy h is given by the well knowndiscrete choice model or ‘Gibbs’ probabilities13

nht = exp(βUh,t−1)

Zt−1, Zt−1 =

H∑h=1

exp(βUh,t−1), (23)

where Zt−1 is a normalization factor in order for the fractionsnht to add up to 1. The crucial feature of (23) is that the higherthe fitness of trading strategy h, the more traders will selectstrategy h. The parameter β in (23) is called the intensityof choice, measuring how sensitive the mass of traders is toselecting the optimal prediction strategy. The intensity ofchoice β is inversely related to the variance of the noise termsεht . The extreme case β = 0 corresponds to the case ofinfinite variance noise, so that differences in fitness cannotbe observed and all fractions (23) will be fixed over time andequal to 1/H . The other extreme case β = ∞ correspondsto the case without noise, so that the deterministic part of thefitness can be observed perfectly and in each period, all traderschoose the optimal forecast. An increase in the intensity ofchoice β represents an increase in the degree of rationalitywith respect to evolutionary selection of trading strategies.The timing of the coupling between the market equilibriumequation (7) or (21) and the evolutionary selection of strategies(23) is crucial. The market equilibrium price pt in (7) dependsupon the fractionsnht . The notation in (23) stresses the fact thatthese fractions nht depend upon past fitnessesUh,t−1, which inturn depend upon past pricespt−1 and dividends yt−1 in periodst − 1 and further in the past as will be seen below. After theequilibrium price pt has been revealed by the market, it willbe used in evolutionary updating of beliefs and determiningthe new fractions nh,t+1. These new fractions nh,t+1 will thendetermine a new equilibrium price pt+1, etc. In the ABS,market equilibrium prices and fractions of different tradingstrategies thus coevolve over time.

A natural candidate for evolutionary fitness is accumulatedrealized profits, as given by

Uht = (pt+yt−Rpt−1)Eh,t−1[pt + yt − Rpt−1]

aσ 2−Ch+wUh,t−1

(24)where R = 1 + r is the gross risk free rate of return, Chrepresents an average per period cost of obtaining forecastingstrategy h and 0 � w � 1 is a memory parameter measuringhow fast past realized fitness is discounted for strategyselection. The cost Ch for obtaining forecasting strategy hwill be zero for simple, habitual rule-of-thumb forecastingrules, but may be positive for more sophisticated forecastingstrategies. For example, costs for forecasting strategies basedupon economic fundamentals may be positive representinginvestors’ effort for information gathering, whereas costs fortechnical trading rules may be (close to) zero. The first term in(24) represents last period’s realized profit of type h given bythe realized excess return of the risky asset over the risk free

13 See Manski and McFadden (1981) and Anderson, de Palma and Thisse(1993) for extensive discussion of discrete choice models and their applicationsin economics.

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asset times the demand for the risky asset by traders of typeh. In the extreme case with no memory, i.e. w = 0, fitnessUht equals net realized profit in the previous period, whereasin the other extreme case with infinite memory, i.e. w = 1,fitnessUht equals total wealth as given by accumulated realizedprofits over the entire past. In the intermediate case, the weightgiven to past realized profits decreases exponentially with time.It will be useful to compute the realized excess return Rt indeviations from the fundamental to obtain

Rt = pt + yt − Rpt−1 = xt + p∗t + yt − Rxt−1 − Rp∗

t−1

= xt − Rxt−1 + p∗t + yt − Et−1[p∗

t + yt ]

+ Et−1[p∗t + yt ] − Rp∗

t−1

≡ xt − Rxt−1 + δt , (25)

where we used that Et−1[p∗t + yt ] − Rp∗

t−1 = 0 since thefundamental p∗

t satisfies the market equilibrium equation (8),and δt ≡ p∗

t + yt − Et−1[p∗t + yt ] is a martingale difference

sequence (MDS). The random term δt enters because thedividend process is stochastic, and thus represents intrinsicuncertainty about economic fundamentals14. According to thedecomposition (25) excess return consists of a conventionalEMH term δt and an additional speculative term xt −Rxt−1 ofthe ABS theory. Our ABS theory thus allows for the possibilityof excess volatility. The extra term is zero if either xt ≡ 0,that is prices equal their fundamental value, or if xt = Rxt−1,that is when prices follow a RE bubble solution. The ABStheory predicts excess volatility in periods when asset pricesgrow faster or slower than the risk free rate of return, or whenprices switch between a temporary bubble solution and thefundamental.

Fitness can now be rewritten in deviations from thefundamental as

Uht = (xt − Rxt−1 + δt )

(fh,t−1 − Rxt−1

aσ 2

)− Ch + wUh,t−1.

(26)

Risk adjusted profits as fitness measure

Although realized net profits are a natural candidate forevolutionary fitness, this fitness measure does not take intoaccount the risk taken at the moment of the investment decision.In fact, given that investors are risk averse mean-variancemaximizers maximizing their expected utility from wealth (4),another natural candidate for fitness is the risk adjusted profit.Using the notation Rt = pt + yt − Rpt−1 for realized excessreturn, the realized risk adjusted profit for strategy h in periodt is given by

πht = Rtzh,t−1 − a

2σ 2z2

h,t−1, (27)

where zh,t−1 = Eh,t−1[Rt ]/(aσ 2) is the demand by trader typeh as in (5). Notice that maximizing expected utility fromwealth in (4) is equivalent to maximizing expected utility fromprofits in (27). A risk adjusted fitness measure based on (27)

14 In the special case of an IID dividend process yt = y + εt we simply haveδt = εt .

is thus consistent with the investors’ demand function derivedfrom mean-variance maximization of expected wealth. Thefitness measure (24) based upon realized profits does not takeinto account the variance term in (27) capturing the investors’risk taken before obtaining that profit. On the other hand, inreal markets realized net profits or accumulated wealth maybe what investors care about most, and the non-risk adjustedfitness measure (24) may thus be practically important.

The expression for risk adjusted profit fitness can besimplified and turns out to be equivalent, up to a constantfactor, to minus squared prediction errors. In order to seethis, we will subtract off the realized risk adjusted profit πRtobtained by rational (perfect foresight) traders from (27). Therisk adjusted profit πRt by rational agents is given by

πRt = Rt Rtaσ 2

− a

2σ 2 R

2t

a2σ 4= R2

t

2aσ 2. (28)

Since πRt is independent of h, subtracting this term from (27)will not affect the maximization of expected utility by tradertype h. Notice also that subtracting this term from (27) will notaffect the fractionsnht of trader typeh, since the discrete choiceprobabilities (23) are independent of the level of the fitness.Using zh,t−1 = Eh,t−1[Rt ]/(aσ 2) a simple computation showsthat

πht − πRt = − 1

2aσ 2(Rt − Eh,t−1[Rt ])

2

= − 1

2aσ 2(pt − Eh,t−1[pt ] + δy,t )

2

= − 1

2aσ 2(xt − Eh,t−1[xt ] + δt )

2, (29)

where δy,t = yt − Et−1[yt ] and δt = p∗t + yt − Et−1[p∗

t + yt ]are both MDS sequences15. The fitness measure risk adjustedprofit is thus, up to a constant factor and an MDS sequence,equivalent to minus squared forecasting errors. The riskadjusted fitness measure is now formally defined as

Vht = − 1

2aσ 2(pt −Eh,t−1[pt ] + δy,t )

2 −Ch +wVh,t−1, (30)

or in deviations from the fundamental

Vht = − 1

2aσ 2(xt − Eh,t−1[xt ] + δt )

2 − Ch + wVh,t−1. (31)

The random term δt or δy,t enters because the dividend processis stochastic, and thus again represents intrinsic uncertaintyabout economic fundamentals.

4. Some simple examplesThis section presents four simple, but typical examples of ABS.The first three subsections discuss the most important featuresof Brock and Hommes (1997b, 1998, 1999), with realizedprofits as the fitness measure. The fourth subsection discussesa modified ABS by Gaunersdorfer and Hommes (2000), with

15 In the special case of an IID dividend process yt = y+εt and correspondingconstant fundamental price p∗ = y/r , we have δy,t = δt = εt .

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evolutionary fitness given by risk adjusted profits conditionedupon deviations from the fundamental. Time series propertiesof the latter example will be compared to 40 years of S&P 500data.

BH present a number of simple, but typical, examples ofthe evolutionary dynamics in adaptive belief systems with two,three or four competing linear forecasting rules (16), wherethe parameter gh represents a perceived trend in prices and theparameter bh represents a perceived upward or downward bias.The ABS then becomes (in deviations from the fundamental):

(1 + r)xt =H∑h=1

nht (ghxt−1 + bh) + εt (32)

nh,t = exp(βUh,t−1)∑Hh=1 exp(βUh,t−1)

(33)

Uh,t−1 = (xt−1 − Rxt−2 + δt−1)

(ghxt−3 + bh − Rxt−2

aσ 2

)

+ wUh,t−2 − Ch, (34)

where the noise term εt represents the model approximationerror and δt−1 represents uncertainty about economicfundamentals as before. In order to keep the analysis of thedynamical behaviour tractable, BH have mainly focused on thecase where the memory parameterw = 0, so that evolutionaryfitness is given by last period’s realized profit. In subsection4.3 we will discuss the role of the memory parameter w. Webriefly discuss three important cases. A common feature of allexamples is that, as the intensity of choice to switch predictionor trading strategies increases, the fundamental steady statebecomes locally unstable and non-fundamental steady states,cycles or even chaos arise.

4.1. Fundamentalists with positive information costsversus trend followers

The simplest example of an ABS only has two trader types,with forecasting rules

f1t = 0 fundamentalists (35)

f2t = gxt−1, g > 0, trend followers (36)

that is, the first type are fundamentalists predicting that theprice will equal its fundamental value (or equivalently thatthe deviation will be zero) and the second type are pure trendfollowers predicting that prices will rise (or fall) by a constantrate. In this example, the fundamentalists have to pay a fixedper period positive cost C1 for information gathering; in allother examples discussed below information costs will be setto zero for all trader types.

For small values of the trend parameter, 0 � g < 1 + r ,the fundamental steady state is always stable. Only forsufficiently high trend parameters, g > 1 + r , can trendfollowers destabilize the system. For trend parameter, 1 + r <g < (1+r)2 the dynamic behaviour of the evolutionary systemdepends upon the intensity of choice to switch between the

two trading strategies16. For low values of the intensity ofchoice, the fundamental steady state will be stable. As theintensity of choice increases, the fundamental steady statebecomes unstable due to a pitchfork bifurcation in which twoadditional non-fundamental steady states −x∗ < 0 < x∗ arecreated. The evolutionary ABS may converge to the positivenon-fundamental steady state, to the negative non-fundamentalsteady state, or, in the presence of noise, switch back andforth between the high and the low steady state. As theintensity of choice increases further, the two non-fundamentalsteady states also become unstable, and limit cycles or evenstrange attractors can arise around each of the (unstable) non-fundamental steady states. The evolutionary ABS may cyclearound the positive non-fundamental steady state, cycle aroundthe negative non-fundamental steady state or, driven by thenoise, switch back and forth between cycles around the highand the low steady state.

This example shows that, in the presence of informationcosts and with zero memory, when the intensity of choice inevolutionary switching is high fundamentalists can not driveout pure trend followers and persistent deviations from thefundamental price may occur. Brock and Hommes (1999)show that this result also holds when the memory in the fitnessmeasure increases. In fact, an increase in the memory of theevolutionary fitness leads to bifurcation routes very similar tobifurcation routes due to an increase in the intensity of choice.

It is sometimes argued that fundamentalists are not rationalsince they do not take into account the presence of othertrader types. Let us therefore briefly discuss the case ofperfect foresight versus trend followers, that is, the casewhen the fundamentalists forecasting rule (35) is replacedby a perfect foresight rule f1t = xt+1. Brock and Hommes(1998, p 1247, lemma 1) show that in this case the firstbifurcation is the same, that is, as the intensity of choiceincreases two non-fundamental steady states are created dueto a pitchfork bifurcation. Although examples with perfectforesight certainly have theoretical appeal, there are twofundamental reasons arguing against perfect foresight in aheterogeneous world. The first is a methodological reason,since with one type having perfect foresight a temporaryequilibrium model with heterogeneous beliefs as in (32), (33)and (34) becomes an implicitly defined dynamical system withxt on the left-hand side and xt+1 and e.g. xt−1 on the right-handside. Typically such implicitly defined evolutionary systemscannot be solved explicitly and often they are not even welldefined17. The second and perhaps more important reason isthat rational expectations or perfect foresight assumes perfectknowledge of the beliefs of all other trader types, which seemsat odds with reality. It seems more reasonable and closer tofinancial practice to focus on examples with fundamentalisttraders.16 For g > (1 + r)2 the system may become globally unstable and prices maydiverge to infinity. Imposing a stabilizing force, for example by assuming thattrend followers condition their rule upon deviations from the fundamental asin subsection 4.4, leads to a bounded system again, possibly with cycles oreven chaotic fluctuations.17 Brock and Hommes (1997a) consider an evolutionary cobweb model withrational expectations at positive information costs versus freely available naiveexpectations. This example can be solved explicitly and exhibits bifurcationroutes to cycles and chaos similar to the ABS presented here.

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4.2. Fundamentalists versus opposite biases

The second example of an ABS is an example with three tradertypes, with forecasting rules

f1t = 0 fundamentalists (37)

f2t = b b > 0, positive bias (optimists) (38)

f3t = −b − b < 0, negative bias (pessimists). (39)

The first type are fundamentalists again, but this time therewill be no information costs. The second and third types havea purely biased belief, expecting a constant price above orrespectively below the fundamental price.

For low values of the intensity of choice, the fundamentalsteady state is stable. As the intensity of choice increasesthe fundamental steady state becomes unstable due to a Hopfbifurcation and the dynamics of the ABS is characterized bycycles around the unstable steady state. This example showsthat when memory is zero, even when there are no informationcosts for fundamentalists, they cannot drive out other tradertypes with opposite biased beliefs. In the evolutionaryABS with high intensity of choice, fundamentalists andbiased traders coexist with fractions varying over time andprices cycling around the unstable fundamental steady state.Moreover, Brock and Hommes (1998, p 1259, lemma 9)show that as the intensity of choice tends to infinity the ABSconverges to a (globally) stable cycle of period 4. Averageprofits along this 4-cycle are equal for all three trader types.Hence, if the initial wealth is equal for all three types, then inthis evolutionary system in the long run accumulated wealthwill be equal for all three types. This example suggests thatthe Friedman argument that smart-fundamental traders willautomatically drive out simple habitual rule of speculativetraders should be considered with care.

In this example with three trader types, cycles can occurbut chaos does not arise18. Therefore, even in the presenceof (small) noise, price fluctuations will be fairly regular andtherefore returns will be predictable. This predictability willdisappear however when we combine trend following withbiased beliefs.

4.3. Fundamentalists versus trend and bias

The third example of an ABS is an example with four tradertypes, with linear forecasting rules (16) with parameters g1 =0, b1 = 0; g2 = 0.9, b2 = 0.2; g3 = 0.9, b3 = −0.2and g4 = 1 + r = 1.01, b4 = 0. The first type arefundamentalists again, without information costs, and the otherthree types follow a simple linear forecasting rule with onelag. For low values of the intensity of choice, the fundamentalsteady state is stable. As the intensity of choice increases, asin the previous three type example, the fundamental steadystate becomes unstable due to a Hopf bifurcation and a stableinvariant circle around the unstable fundamental steady statearises, with periodic or quasi-periodic fluctuations. As theintensity of choice further increases, the invariant circle breaksinto a strange attractor with chaotic fluctuations.

18 This may be seen from the plot of the largest Lyapunov exponent in Brockand Hommes (1998, p 1261, figure 9(b)), which is always non-positive.

This example shows that when memory is zero, evenwhen there are no information costs for fundamentalists, theycannot drive out other simple trader types and fail to stabilizeprice fluctuations towards the fundamental value. As in thethree type case, the opposite biases create cyclic behaviourbut apparently the additional trend parameters turn thesecycles into chaotic fluctuations. In the evolutionary ABSfundamentalists and chartists coexist with fractions varyingover time and prices moving chaotically around the unstablefundamental steady state. The corresponding strange attractor,with the parameter values given by r = 0.01, β = 90.5,w = 0and Ch = 0 for all 1 � h � 4, was already shown in figure 1.Figure 2 shows a chaotic as well as a noisy chaotic time series.

The (noisy) chaotic price fluctuations are characterized byan irregular switching between phases of close-to-the-EMH-fundamental-price fluctuations, phases of ‘optimism’ withprices following an upward trend, and phases of ‘pessimism’,with (small) sudden market crashes. Recall from subsection3.1 that the asset pricing model with homogeneous beliefs,in addition to the benchmark fundamental price, has rationalbubble solutions as in (12). One might say that the ABS pricesare characterized by an evolutionary switching between thefundamental value and these temporary speculative bubbles.

Brock and Hommes (1997a, 1998) show that for a highintensity of choice, the ABS system is close to having ahomoclinic orbit, Poincare’s classical notion and key feature ofchaotic systems. In the purely deterministic chaotic case, thetiming and the direction of the temporary bubbles seem hard topredict (see figure 2(a)). However, once a bubble has started,in the deterministic case, the length of the bubble seems to bepredictable in most of the cases. In the presence of noise, asin figure 2(b), the timing, the direction and the length of thebubble all seem hard to predict. In order to investigate this(un)predictability issue further, we employ a so called nearestneighbour forecasting method to predict the returns, at lags 1to 20 for the purely chaotic as well as for several noisy chaotictime series, as illustrated in figure 319.

Nearest neighbour forecasting looks for past patternsclose to the most recent pattern, and then yields as theprediction the average value following all nearby past patterns.It follows essentially from Takens’ embedding theoremthat this method yields good forecasts for deterministicchaotic systems20. Figure 3 shows that as the noise levelincreases, the forecasting performance of the nearest neighbourmethod quickly deteriorates. Hence, in our simple nonlinearevolutionary ABS with noise it is hard to make good forecastsof future returns. Our simple nonlinear ABS with small noisethus captures some of the intrinsic unpredictability of assetreturns also present in real markets.

Finally, let us briefly discuss the issue of memory inthe fitness measure. Figure 4 shows bifurcation diagramsof the deterministic ABS as well as the noisy ABS, withrespect to the memory parameter w, 0 � w � 1. Theparameters are as before, except for the intensity of choicewhich has been chosen smaller, β = 40, so that in the

19 I would like to thank Sebastiano Manzan for providing this figure.20 See Takens (1981) and Packard et al (1980). A recent treatment of nonlineartime series analysis and forecasting techniques is Kantz and Schreiber (1997).

158


0.8

0.4

0

–0.4

–0.8

0 120 240 360 480

(a) 1.1

0.5

–0.1

–0.7

–1.3

0 120 240 360 480

(b)

Figure 2. Chaotic (a) and noisy chaotic (b) time series of asset prices in an adaptive belief system with four trader types. Belief parametersare: g1 = 0, b1 = 0; g2 = 0.9, b2 = 0.2; g3 = 0.9, b3 = −0.2 and g4 = 1 + r = 1.01, b4 = 0; other parameters are r = 0.01, β = 90.5,w = 0 and Ch = 0 for all 1 � h � 4.

��

��

��

��

��

��

��

��

��

��

Figure 3. Forecasting errors for the nearest neighbour methodapplied to chaotic returns series as well as noisy chaotic returnsseries, for different noise levels, in ABS with four trader types. Allreturns series have close to zero autocorrelations at all lags. Thebenchmark case of prediction by the mean 0 is represented by thehorizontal line at the normalized prediction error 1. Nearestneighbour forecasting applied to the purely deterministic chaoticseries leads to much smaller forecasting errors (lowest graph). Anoise level of say 10% means that the ratio of the variance of thenoise term εt and the variance of the deterministic price series is1/10. As the noise level slowly increases, the graphs are shiftedupwards. Small dynamic noise thus quickly deteriorates forecastingperformance.

case with zero memory (w = 0) the fundamental steadystate is stable. As the memory parameter increases, thefundamental steady becomes unstable for w ≈ 0.3, due to aHopf bifurcation, and as memory approaches 1 (the case wherefitness is given by accumulated wealth) the dynamics becomesmore complicated, even chaotic. The (long) time series infigure 4 shows that with memory close to 1 speculative bubblesstill occur, although they are less frequent than for smallermemory. In fact, increasing memory yields a bifurcation routeto instability, cycles and chaos similar to the bifurcation routeswith respect to an increase in the intensity of choice. Theintuition behind this result is as follows. Even when the

intensity of choice to switch strategies is low, when memory infitness is large, differences in accumulated profits can becomesufficiently large to cause the majority of traders to switch to atrend following strategy and leading to a (temporary) bubble.More memory thus in general does not stabilize an evolutionarysystem, but may in fact be destabilizing21.

4.4. An example with volatility clustering

From a qualitative viewpoint, the price fluctuations in thesimple examples of the nonlinear noisy ABS are similarto observed fluctuations in real markets. But do these(chaotic) fluctuations explain a significant part of stock pricefluctuations? Brock and Hommes (1997b) calibrated the ABSof the previous subsection to ten years of monthly IBM pricesand returns and found that the autocorrelations of both the(noisy) chaotic returns and squared returns had no significantautocorrelations, similarly for the monthly IBM returns andsquared returns. In this section we discuss a modified ABSof Gaunersdorfer and Hommes (2000) exhibiting volatilityclustering and fat tails. There are three modifications comparedwith the BH-models discussed before: (i) the evolutionaryfitness measure is given by risk adjusted realized profits, asopposed to (non-risk adjusted) realized profits; (ii) technicaltraders condition their rules upon market fundamentals, whichis reflected in the second stage of the updating scheme forfractions as discussed below; (iii) we allow for one extra timelag in the trend following forecasting rules, leading to a 4D (asopposed to 3D) system in the simplest zero memory case.

Let there be two types of traders, with forecasting rules

pe1,t+1 = f1t = p∗ + v(pt−1 − p∗), 0 � v � 1,

fundamentalists (40)

pe2,t+1 = f2t = pt−1 + g(pt−1 − pt−2), g � 0,

trend extrapolators. (41)

21 Theoretically, it remains an open question whether in the infinite memorycase w = 1, so that fitness equals accumulated wealth, fundamentalists willbe able to stabilize the price towards its fundamental value. The time seriesin figure 4(c) shows that for w ≈ 0.99 say, in the presence of small noisethe system is unstable and fundamentalists cannot drive out all other types.Hence, even when traders discount past realized profits only weakly, trendfollowing trading strategies may survive evolutionary competition.

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1.8

0.9

0

–0.9

–1.8

1.4

0.6

–0.2

–1

–1.8

0 0.4 0.6 0.8 1 00.2 0.4 0.6 0.8 10.2

1.7

0.8

–0.1

–1

–1.9

0 500 1000 1500 2000

Figure 4. Bifurcation diagrams with respect to the memory parameter w, without noise (a) and with small noise (b); other parameters are asin figure 2. Time series for memory parameter w = 0.99 in ABS with small noise (c).

Trader type 1 are fundamentalists, believing that tomorrow’sprice will move in the direction of the fundamental price p∗ bya factor v. We will concentrate here on the special case v = 1,so that

f1t = pt−1, EMH believer (42)

and we will refer to this type as an EMH believer becausethe naive forecast is consistent with a random walk for prices.Trader type 2 are simple trend extrapolators, extrapolating thelatest observed price change, so that the forecasting rule nowincludes two time lags. Market equilibrium (7) in a world withfundamentalists and chartists as in (40)–(41), with commonexpectations on IID dividends Et [yt+1] = y, becomes

(1 + r)pt = n1t (p∗ + v(pt−1 − p∗))

+ n2t (pt−1 + g(pt−1 − pt−2)) + y + εt , (43)

where n1t and n2t represent the fractions of fundamentalistsand chartists respectively and εt is an IID random variablerepresenting model approximation errors.

Beliefs will be updated by conditionally evolutionaryforces. The basic idea is that fractions are updatedaccording to past fitness, conditioned upon the deviation ofactual prices from the fundamental price. The evolutionarycompetitive part of the updating scheme follows the BH-framework with risk adjusted profits as the fitness measure; theadditional conditioning upon deviations from the fundamentalis motivated by the approach taken in the Santa Fe artificialstock market in Arthur et al (1997) and LeBaron et al (1999).Using (30) with zero costs and zero memory, the evolutionarypart of the updating of fractions yields the discrete choice

160


probabilities

nht = exp

[− β

2aσ 2(pt−1 − fh,t−2 + δt )

2

]/Zt−1,

h = 1, 2

(44)

where Zt−1 is again the normalization factor such that thefractions add up to one. In the second step of updating offractions, the conditioning on deviations from the fundamentalby the technical traders is modelled as

n2t = n2t exp[−(pt−1 − p∗)2/α], α > 0 (45)

n1t = 1 − n2t . (46)

The conditioning upon fundamentals part of the updatingscheme may be seen as a stabilizing transversality conditionin a heterogeneous world. Recall that in the perfectly rationalworld of subsection 3.1 the transversality condition excludesthe rational expectations explosive bubble solutions (12).According to (45) the fraction of technical traders decreasesmore, the further prices deviate from their fundamentalvalue p∗. As long as prices are close to the fundamental,updating of fractions will be almost completely determined byevolutionary fitness (44), but when prices move far away fromthe fundamental, the correction term exp[−(pt−1 − p∗)2/α]in (45) becomes small. The majority of technical analyststhus believes that temporary speculative bubbles may arise butthat these bubbles cannot last forever and that at some pointa price correction towards the fundamental price will occur.The condition (45) may thus be seen as a weakening of thetransversality condition in a perfectly rational world, allowingfor temporary speculative bubbles.

The noisy conditional evolutionary ABS with fundamen-talists versus chartists is given by (40)–(41) and (43)–(46). Bysubstituting all equations into (43) a fourth order nonlinearstochastic difference equation in prices pt is obtained. It turnsout that this nonlinear evolutionary system exhibits periodic aswell as chaotic fluctuations of asset prices and returns; a de-tailed mathematical analysis of the bifurcation routes to strangeattractors and coexisting attractors is given in Gaunersdorfer,Hommes and Wagener (2000). Here we focus on one simple,but typical example with EMH believers, i.e. v = 1, versustrend followers. Figure 5 compares time series properties ofthe noisy ABS with 40 years of S&P 500 data22.

The price series in the top panels are of course quitedifferent, since S&P 500 is non-stationary and stronglyincreasing, whereas the ABS is a stationary model23. Pricesin the ABS system are highly persistent however, and are in

22 These results were obtained mainly by ‘trial and error’ numericalsimulations. Strong volatility clustering was obtained especially for v = 1or v very close to 1, and seems to be fairly robust with respect to the otherparameter values. For smaller values of v the volatility clustering becomesweaker. This may be due to the fact that for smaller values of v prices returnfairly quickly back to the constant fundamental, and volatility dies out quickly.23 By replacing our IID dividend process by a non-stationary dividend process,e.g. by a geometric random walk, prices in the ABS will also rapidly increase,similar to the S&P 500 series. We intend to study such non-stationary ABS infuture work.

fact close to having a unit root. In the S&P 500 returns seriesthe October 1987 crash and the two days thereafter have beenexcluded24. The ABS returns series ranges from −0.27 to+0.29, which is larger than for the S&P 500, especially whenthe crash is excluded. The ABS returns series exhibits fat tails,with kurtosis coefficient k = 5.37 versus k = 8.51 for the S&P500 returns, and strong volatility clustering. We estimated aGARCH(1, 1) model of the form

Rt = c1 + δt , δt ∼ N(0, σ 2t ) (47)

σ 2t = c2 + ρ1δ

2t−1 + ρ2σ

2t−1 (48)

on both returns series. The estimated parameters are ρ1 =0.068 and ρ2 = 0.929, with ρ1 + ρ2 = 0.997, for S&P 500returns and ρ1 = 0.071 and ρ2 = 0.914, with ρ1 + ρ2 =0.985, for the ABS returns. Figure 5(e), (f ) shows that theautocorrelations of the returns, squared returns and the absolutereturns of the ABS-model series are very similar to thoseof S&P 500, with (almost) no significant autocorrelations ofreturns and slowly decaying autocorrelations of squared andabsolute returns. Although the ABS system considered hereis a nonlinear dynamic system with only four lags, it exhibitslong memory with long range autocorrelations. The bottompanels show a scaling law of the form

ρj = k

jα(49)

fitted to the autocorrelations of absolute returns, for lags5 � j � 100. The estimated scaling exponents are 0.38for S&P 500 and 0.37 for the ABS; see Mantegna and Stanley(2000) for a survey on the role of scaling laws in financial timeseries. Our simple ABS thus exhibits a number of importantstylized facts of 40 years of S&P 500 returns data.

5. Asset pricing experimentsMuch computational and theoretical work on expectationformation in heterogeneous agent systems has been done in thepast decade, but it is hard to test the heterogeneous expectationshypothesis empirically and to infer the expectations hypothesisfrom economic or financial data. Survey data research, asfor example in Shiller (1989) on stock market expectations,yields useful insight on expectation formation but also has itslimitations, for example because of unknown and changingunderlying economic fundamentals. Controlled laboratoryexperiments seem to be well suited to investigate expectationformation and forecasting behaviour in particular situations.As noted e.g. by Sunder (1995), it is remarkable that, despitean explosion of interest in experimental economics, relativelyfew contributions have focused on expectation formation andlearning in dynamic experimental markets with expectationsfeedback. An exception is the well known ‘bubble experiment’by Smith, Suchanek and Williams (1988) in an experimentalasset market. This study cannot be viewed however as pure

24 The returns for these days were about −0.20, +0.05 and +0.09. In particular,the crash affects the autocorrelations of squared returns, which drop to smallvalues of 0.03 or less for all lags k � 10 when the crash is included.

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Figure 5. S&P 500 data (left-hand panels) compared to ABS simulated data (right-hand panels), with parameters: r = 0.01, y = 3,p∗ = 300, v = 1, g = 1, β = 5, α = 50, δt ≡ 0 and εt normally distributed with σε = 11. Prices (top panels (a), (b)), returns (second panels(c), (d)) defined as relative price changes, ACFs of returns, squared returns and absolute returns (third panels (e), (f )) and ACFs of absolutereturns with fitted scaling laws (bottom panels (g), (h)).

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experimental testing of the expectations hypothesis, everythingelse being held constant, because dynamic market equilibriumis affected not only by expectations feedback but also by othertypes of human behaviour, such as trading behaviour.

This section discusses some work in progress, jointlywith my colleagues Joep Sonnemans, Jan Tuinstra and Henkvan de Velden from CeNDEF, testing expectation formationin the CREED experimental laboratory at the University ofAmsterdam. Before presenting some first results from theasset pricing framework we will briefly discuss some generalconclusions from a series of three related experimental papers,Hommes et al (1999a, 2000a) and Sonnemans et al (1999),testing for expectation formation in a different but relatedframework, the cobweb model. The cobweb or ‘hog cycle’model is perhaps the simplest dynamic economic expectationsfeedback system. The cobweb model describes a singlecommodity market with a production lag, so that suppliers haveto form price expectations one period ahead. A convenientfeature of the cobweb model is that it has a unique rationalexpectations equilibrium, namely the steady-state price atwhich demand and supply intersect. Market equilibriumequations were controlled and fixed during the experiment(although they were subject to unexpected exogenous shocksand/or noise). Subjects were asked to predict prices and theirearnings were inversely related to their quadratic forecastingerrors. Price forecasts fed into the (unknown) supplyfunction, and the realized market price was then determinedby equilibrium between aggregate supply and demand. Themarket price realizations only depend upon subjects’ priceexpectations. Demand and supply were chosen such that undernaive price expectations, the RE steady state is locally unstableand prices diverge away from the RE steady state and convergeto a 2-cycle, with ‘systematic’ forecasting errors. An importantmotivation was whether individuals in the experiment wouldbe able to learn from their forecasting errors and coordinateon the RE steady state. For the cobweb experiments, the mainconclusions can be summarized as follows:

1. In most experimental treatments the sample mean ofrealized market prices is close to the RE price. The nullhypothesis that the sample mean of realized market pricesequals the RE steady state price can not be rejected. Inthis sense, the RE forecast is on average correct;

2. In all experimental treatments realized market pricesexhibit significant excess volatility, that is, the nullhypothesis that the sample variance of realized marketprices is smaller than or equal to the RE-variance isstrongly rejected;

3. Realized market prices are characterized by irregularfluctuations around the mean and exhibit hardly any linearpredictability, since sample autocorrelations are typicallyinsignificant.

To summarize the cobweb expectations experiments in onesentence, the (unstable) experimental cobweb economy seemsto be consistent with heterogeneous boundedly rational agents,using a variety of boundedly rational forecasting rules creatingexcess price volatility around the RE mean.

Figure 6 shows the outcome of some first asset pricingexperiments with expectations feedback, from Hommes et al(2000b)25.

Participants were asked to predict the price p, 0 � p �100, of a risky asset and earnings were inversely related toforecasting accuracy, or equivalently to risk adjusted profits.In contrast to the cobweb experiments, the experimental assetpricing framework is complicated by the existence of multipleRE equilibria, since the asset pricing model has RE ‘bubblesolutions’ growing at the risk free rate of return r . These bubblesolutions are perfect foresight solutions, where forecastingerrors are zero (or equal to the noise term). If participants in theexperiments are motivated by minimizing forecasting errors,both these bubble solutions and the fundamental solution p∗

are equally attractive. To avoid the existence of an unboundedperfect foresight bubble solution, we introduced a stabilizingfundamental type F robot trader in the experimental setup.In the presence of a fundamental robot trader, the (constant)fundamental price is the only perfect foresight solution. Ourexperimental setup thus represents a situation where at leastsome of the traders in the market know and believe in thefundamental price, and the number (i.e. the weight) of thefundamentalists increases when prices move away from thefundamental.

The fundamental type F always expects the fundamentalprice to prevail, i.e. EFt(pt+1) = p∗. In the asset pricingexperiments the (unknown) market equilibrium is given by

(1 + r)pt =H∑h=1

nhtEht (pt+1) + nFtp∗ + y + εt , (50)

where nht , 1 � h � H , are the weights (or fractions) of theparticipants upon realized market equilibrium prices and nFtis the weight (or fraction) of the fundamental robot traders inthe market. In our experimental setup H = 6 and the weightsnht , 1 � h � H , of the participants will decrease when pricesmove away from the fundamental price p∗. More precisely,these weights will be given by

nht = exp(−α|pt−1 − p∗|)H

,

1 � h � H, α = 0.005 (51)

and the weight of the fundamentalists by

nFt = 1 −H∑h=1

nht . (52)

Hence, when prices move away from the fundamental p∗, theweights nht , 1 � h � H , of the participants upon realizedmarket equilibrium price pt decrease and the weight nFt ofthe fundamental robot trader increases. The fundamentalrobot trader in our experimental setup thus stabilizes possiblebubble solutions. Notice however, that all weights nht of theparticipants are equal, and close to 1/H as long as realizedmarket prices are close to the fundamental price. In theexperiments, mean dividend y = 3 and the risk free rate of

25 I would like to thank Henk van de Velden for providing these figures.

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(a) (b)

(c) (d)

Figure 6. Asset pricing experiments. A stable market (a) and a market with bubbles (b). Individual expectations are included in the lowerpanels (c), (d).

return r = 0.05 were common knowledge. All participantsthus had sufficient information to compute the fundamentalprice p∗ = y/r = 60. The noise εt was normally distributedwith standard deviation σε = 0.5, so that in a perfectly rationalworld prices would fluctuate between 59 and 61 most of thetime.

In the first experiment (figure 6(a)) the market is stable andconverges to a value close to the RE fundamental. The realizedmarket price is however always below the RE fundamental, sothat the market is in fact undervalued. In the second experiment(figure 6(b)) participants coordinate on a speculative bubblesolution. The first speculative bubble lasts 13 periods, butreverses and becomes a stable oscillation due to the presence ofthe fundamental robot trader. The first maximum occurs after13 time periods; thereafter a stable oscillation arises, with local

maxima at periods 23, 29, 35 and 41. The bottom panels showthe realized market price as well as the individual forecasts byall six participants in the experiments. In the stable case (figure6(c)) participants quickly coordinate on a price below but fairlyclose to the fundamental price p∗ = 60. In the unstable case,all participants quickly coordinate on a speculative bubble.However, as time goes on, the coordination becomes weaker,and the bubble seems to stabilize.

It should be noted that the fundamental price in theseexperiments is constant and thus as simple as possible. Evenin this simplest setting, (temporary) bubbles arise but theyseem to stabilize toward the end of the experiment. Animportant question for future work is what will happen whenthe fundamental robot trader is not present in the market. Willthe speculative bubbles stabilize in the absence of fundamental

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robot traders? In future experiments we will also focus on othercontrolled fundamentals, for example a constant fundamentalwith larger noise levels or, what seems to be important for realstock markets, a time varying, growing fundamental.

6. Future perspectiveIs a significant part of changes in stock prices driven by‘Keynesian animal spirits’? For many decades, this questionhas lead to heavy debates among economic academics aswell as financial practitioners. In the evolutionary adaptivebelief systems discussed here, price changes are explainedby a combination of economic fundamentals and ‘marketpsychology’. Negative economic ‘news’ (e.g. on inflationor interest rates) may act as a trigger event for a declinein stock prices, which may become reinforced by investorssentiment and evolutionary forces. Price movements are drivenby an interaction of fundamentalism and chartism, the twomost important trading strategies in financial practice. Thenonlinear evolutionary ABS generates a number of importantstylized facts observed in many financial series, such asunpredictability of returns, fat tails, strong volatility clusteringand long memory.

Let us now reflect upon the questions Q1–Q3 raised atthe end of the introduction, which were central to the SFIworkshop.

Can speculators survive evolutionary competition?

Friedman (1953, p 175) has argued that speculators will notsurvive evolutionary competition: ‘People who argue thatspeculation is generally destabilizing seldom realize that this islargely equivalent to saying that speculators lose money, sincespeculation can be destabilizing in general only if speculatorson the average sell when the currency is low in price andbuy when it is high’ [emphasis added]. We have seen thatin the evolutionary ABS technical trading rules do surviveevolutionary competition and can in fact earn profits or attainwealth comparable to profits or wealth of fundamentalists,even when there are no information costs and memory inevolutionary fitness is high. The technical analysts start buyingwhen prices are low, in the early stage of an upward pricetrend which may have been triggered by news about economicfundamentals, and sell as soon as their trading rule detectsthat the trend has reversed when the price is still high. Themain reason why technical trading can survive evolutionarycompetition seems to be the fact that in markets for risky assetsan optimistic or pessimistic mood leads to a self-fulfillingspeculative bubble when the optimism or pessimism is sharedby a large enough group of investors. The theoretical questionexactly which trading strategy generates highest profits orwealth is a difficult problem to deal with in a complex nonlinearheterogeneous world and the exact answer remains open. Oursimple examples and numerical simulations show howeverthat, in a boundedly rational heterogeneous world technicaltrading rules are not necessarily inferior to strategies basedupon economic fundamentals and that a well-timed switchingstrategy between trend following and fundamentalism mayoutperform pure chartists as well as pure fundamentalists.

Is a financial market with heterogeneousadaptive agents efficient?

There seems to be disagreement about exactly what efficiencymeans. But there are at least two important factorsrelated to efficiency. One is sometimes called informationalefficiency, meaning that a market should be difficult toforecast, since otherwise there would be obvious arbitrageopportunities. Our simple nonlinear evolutionary ABS is,at least to some degree, informationally efficient becauseasset returns are fairly unpredictable and have e.g. closeto zero autocorrelations. Employing advanced time seriesmethods such as nearest neighbour forecasting does not leadto very accurate predictions of returns due to the strong noiseamplification in nonlinear evolutionary systems. Nevertheless,figure 3 shows that even in the worst case with the highestnoise level, the prediction error for short time lags is below0.9 which in a real financial market might lead to largeprofit opportunities. It should be emphasized though, thatthese results have been obtained for an extremely simpleform of our ABS, a low-dimensional (3D) version with aconstant fundamental price. Increasing the dimension of theABS, e.g. by introducing more lags into the forecasting rules,and including a more realistic, non-stationary time varyingfundamental will make forecasting of returns much moredifficult and even more sensitive to noise, moving the ABSeven closer to informational efficiency.

A second factor concerning efficiency is sometimesreferred to as allocative efficiency meaning that asset pricesreflect the ‘true’ fundamental value of the underlying asset.In the evolutionary ABS large and persistent deviations fromthe fundamental can occur, possibly triggered by noise andreinforced by evolutionary forces. An evolutionary ABS thusallows for the possibility of allocative inefficiency. Knowledgeabout the true fundamentals is important in this respect. Ifmost traders agree about the ‘true’ fundamental and when it isknown with great precision, deviations from this commonlyshared fundamental seem unlikely. However, in a worldfull of uncertainty, where nobody really knows what exactlythe fundamental is, good news about economic fundamentalsamplified by evolutionary forces may lead to deviations fromthe fundamental and over- or under-valuation of asset prices.

Does heterogeneity create excess volatility?

The ABS theory implies a decomposition of returns into twoterms, one martingale difference sequence part according tothe conventional EMH theory, and an extra speculative termadded by the evolutionary theory. The theory of evolutionaryABS thus can explain excess volatility. In an ABS thephenomenon of volatility clustering occurs endogenously dueto the interaction of heterogeneous traders. In periods of lowvolatility the market is dominated by fundamentalists. Phasesof high volatility may be triggered by news about fundamentalsand may be amplified by technical trading.

Our evolutionary ABS may be seen as stylized modelsfitting in recent work on behavioural economics andbehavioural finance as discussed, for example, in Thaler

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(1994). Our ABS deviate from full rationality and areconsistent with recent empirical findings that technical tradingstrategies may earn above average profits as, for example,found by Brock et al (1992). Our evolutionary ABS mayalso be seen as what Sargent calls an approximate rationalexpectations equilibrium (see, for example, Sargent (1993)and Hommes and Sorger (1998) for extensive discussion ofthis point). Traders are boundedly rational and use relativelysimple strategies. The class of trading rules is disciplinedby evolutionary forces based upon realized profits or wealth.A convenient feature of our theoretical setup is that thebenchmark rational expectations model is nested as a specialcase. This feature gives the model flexibility with respect toexperimental and empirical testing. It is worthwhile noting thatChavas (1999) and Baak (1999) have run empirical tests forheterogeneity in expectations in agricultural data and indeedfind evidence for the presence of boundedly rational traders inthe hog and cattle markets. It may seem even more natural thatheterogeneity and evolutionary switching between differenttrading strategies play an important role in financial markets.Understanding the role of market psychology seems to be acrucial part of understanding the huge changes in stock pricesobserved so frequently these days. But much more insight into‘financial psychology’ is needed, before ‘market sentiment’based policy advice can be given. Theoretical analysis ofstylized evolutionary adaptive market systems, as discussedhere, and its empirical and experimental testing may contributeto providing such insight

AcknowledgmentsI am much indebted to Buz Brock, whose stimulating ideasto a large extent have influenced my way of thinking aboutmarkets. This paper heavily draws from our joint work andmany stimulating discussions over the past seven years. Ialso would like to thank Andrea Gaunersdorfer for manystimulating discussions; some of our recent joint work isincluded in this paper. An earlier version of this paper waspresented at the Santa Fe Institute workshop ‘Beyond Efficiencyand Equilibrium’, 18–20 May 2000. I would like to thankthe organizers of the workshop, Doyne Farmer and JohnGeanakoplos, for the opportunity to present this work, andthe participants of the SFI workshop for many stimulatingdiscussions. Detailed comments by Doyne Farmer and ananonymous referee on an earlier draft have been very helpful.Finally, I would like to thank my colleagues at the Center forNonlinear Dynamics in Economics and Finance (CeNDEF)at the University of Amsterdam, for providing a stimulatingresearch environment to pursue the ideas reflected in this paper.Financial support for this research by an NWO-MaG Pioniergrant is gratefully acknowledged.

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30TH ANNIVERSARY ISSUE 2004 THE JOURNAL OF PORTFOLIO MANAGEMENT 15

The 30th anniversary of The Journal of Portfolio Man-agement is a milestone in the rich intellectual his-tory of modern finance, firmly establishing therelevance of quantitative models and scientific

inquiry in the practice of financial management. One of themost enduring ideas from this intellectual history is the Effi-cient Markets Hypothesis (EMH), a deceptively simplenotion that has become a lightning rod for its disciples andthe proponents of behavioral economics and finance.

In its purest form, the EMH obviates active portfo-lio management, calling into question the very motivationfor portfolio research. It is only fitting that we revisit thisgroundbreaking idea after three very successful decades ofthis Journal.

In this article, I review the current state of the con-troversy surrounding the EMH and propose a new per-spective that reconciles the two opposing schools of thought.The proposed reconciliation, which I call the Adaptive Mar-kets Hypothesis (AMH), is based on an evolutionary approachto economic interactions, as well as some recent research inthe cognitive neurosciences that has been transforming andrevitalizing the intersection of psychology and economics.

Although some of these ideas have not yet been fullyarticulated within a rigorous quantitative framework, longtime students of the EMH and seasoned practitioners willno doubt recognize immediately the possibilities generatedby this new perspective. Only time will tell whether itspotential will be fulfilled.

I begin with a brief review of the classic version ofthe EMH, and then summarize the most significant criti-cisms leveled against it by psychologists and behavioraleconomists. I argue that the sources of this controversy can

ANDREW W. LO is Harris &Harris Group Professor at theMIT Sloan School of Man-agement, and chief scientificofficer at AlphaSimplex Group,LLC, in Cambridge, MA.

The Adaptive Markets HypothesisMarket efficiency from an evolutionary perspective.

Andrew W. Lo

It is illegal to reproduce this article in any format. Copyright 2004.

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be traced back to the very origins of modern neoclassical eco-nomics, and, by considering the sociology and culturalhistory of modern finance, we can develop a better under-standing of how we arrived at the current crossroads for theEMH.

I then turn to the AMH, in which the dynamics ofevolution—competition, mutation, reproduction, and nat-ural selection—determine the efficiency of markets and thewaxing and waning of financial institutions, investmentproducts, and ultimately institutional and individual for-tunes. I conclude by considering some implications of theAMH for portfolio management, and by outlining a researchagenda for formalizing several aspects of the model.1

CLASSICAL EFFICIENT MARKETS HYPOTHESIS

We all know the joke about an economist strollingdown the street with a companion. They come upon a$100 bill lying on the ground. As his companion reachesdown to pick it up, the economist says, “Don’t bother—ifit were a genuine $100 bill, someone would have alreadypicked it up.”

This example of economic logic gone awry is a fairlyaccurate rendition of the Efficient Markets Hypothesis, oneof the most contested propositions in all the social sciences.It is disarmingly simple to state; it has far-reaching conse-quences for academic theories and business practice; and yetis surprisingly resilient to empirical proof or refutation. Evenafter several decades of research and literally thousands of stud-ies, many published in this Journal, economists have not yetreached a consensus about whether markets—particularlyfinancial markets—are in fact efficient.

As with so many of the ideas of modern economics,the origins of the EMH can be traced back to Paul Samuel-son [1965], whose contribution is neatly summarized by histitle, “Proof that Properly Anticipated Prices Fluctuate Ran-domly.” In an informationally efficient market, price changesmust be unforecastable if they are properly anticipated, thatis, if they fully incorporate the information and expectationsof all market participants. Roberts [1967] and Fama [1970]operationalized this hypothesis—summarized in Fama’s well-known description, “prices fully reflect all available infor-mation”—by placing structure on various information setsavailable to market participants.

This concept of informational efficiency has a Zen-like,counterintuitive flavor to it. The more efficient the market,the more random the sequence of price changes generatedby such a market; and the most efficient market of all is amarket in which price changes are completely random andunpredictable. This is not an accident of nature, but is in factthe direct result of many active market participants attempt-ing to profit from their information.

Driven by profit opportunities, an army of investors

pounce on even the smallest informational advantages at theirdisposal. In doing so, they incorporate their information intomarket prices and quickly eliminate the profit opportunitiesthat first motivated their trades. If this occurs instantaneously,which it must in an idealized world of frictionless marketsand costless trading, prices must always fully reflect all avail-able information. Therefore, no profits can be garneredfrom information-based trading because such profits musthave already been captured (the $100 bill on the ground).In mathematical terms, prices follow martingales.

A decade after Samuelson’s [1965] landmark work, hisframework was broadened to accommodate risk-averseinvestors, yielding a neoclassical version of the EMH whereprice changes, properly weighted by aggregate marginal util-ities, must be unforecastable (see, for example, LeRoy [1973];Rubinstein [1976]; and Lucas [1978]). In markets where,according to Lucas [1978], all investors have “rational expec-tations,” prices do fully reflect all available information andmarginal-utility weighted prices follow martingales.

The EMH has been extended in many other directions,to incorporate non-traded assets such as human capital,state-dependent preferences, heterogeneous investors, asym-metric information, and transaction costs. But the generalthrust is the same: Individual investors form expectationsrationally; markets aggregate information efficiently; andequilibrium prices incorporate all available information.

The current version of the EMH can be summarizedcompactly by the “three Ps of Total Investment Manage-ment”: prices, probabilities, and preferences (see Lo [1999]).The three Ps have their origins in one of the most basic andcentral ideas of modern economics, the principle of supplyand demand.

This principle states that the price of any commodityand the quantity traded are determined by the intersectionof supply and demand curves, where the demand curverepresents the schedule of quantities desired by consumersat various prices, and the supply curve represents the sched-ule of quantities producers are willing to supply at variousprices. The intersection of these two curves determines an“equilibrium,” a price-quantity pair that satisfies both con-sumers and producers simultaneously. Any other price-quan-tity pair may serve one group’s interests, but not the other’s.

Even in this simple description of a market, all the ele-ments of modern finance are present. The demand curve isthe aggregation of many individual consumers’ desires, eachderived from optimizing an individual’s preferences, subjectto a budget constraint that depends on prices and other fac-tors (e.g., income, savings requirements, and borrowingcosts). Similarly, the supply curve is the aggregation of manyindividual producers’ outputs, each derived from optimiz-ing an entrepreneur’s preferences, subject to a resource con-straint that also depends on prices and other factors (e.g., costsof materials, wages, and trade credit). And probabilities affect

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both consumers and producers as they formulate their con-sumption and production plans through time and in the faceof uncertainty—uncertain income, uncertain costs, anduncertain business conditions.

It is the interactions among prices, preferences, andprobabilities that give modern financial economics its rich-ness and depth. Formal models of financial asset prices suchas Leroy [1973], Merton [1973], Rubinstein [1976], Lucas[1978], and Breeden [1979] show precisely how the threePs simultaneously determine a “general equilibrium” inwhich demand equals supply across all markets in an uncer-tain world where individuals and corporations act rationallyto optimize their own welfare. The three Ps enter into anyeconomic decision under uncertainty. It may be argued thatthey are fundamental to all forms of decision-making.

BEHAVIORAL CRITIQUES

The three Ps of Total Investment Management yieldquite specific theoretical and empirical implications thathave been tested over the years. The early tests focused pri-marily on whether prices of certain financial assets do fullyreflect various types of information, and several tests have alsoconsidered the characteristics of probabilities implicit inasset prices (see, for example, Cootner [1964] and Lo [1997]).But the most enduring critiques of the EMH revolve aroundthe preferences and behavior of market participants.

The standard approach to modeling preferences is toassert that investors optimize additive time-separable expectedutility functions from certain parametric families, e.g., con-stant relative risk aversion. Psychologists and experimentaleconomists have documented a number of departures fromthis paradigm, though, in the form of specific behavioral biasesthat are endemic in human decision-making under uncer-tainty, and several of these lead to undesirable outcomes foran individual’s economic welfare. They include: overconfi-dence (Fischoff and Slovic [1980]; Barber and Odean [2001];Gervais and Odean [2001]), overreaction (De Bondt andThaler [1986]), loss aversion (Kahneman and Tversky [1979];Shefrin and Statman [1985]; Odean [1998]), herding (Huber-man and Regev [2001]), psychological accounting (Tverskyand Kahneman [1981]), miscalibration of probabilities (Licht-enstein, Fischoff, and Phillips [1982]), hyperbolic discount-ing (Laibson [1997]), and regret (Bell [1982]; Clarke, Krase,and Statman [1994]). These critics of the EMH argue thatinvestors are often if not always irrational, exhibiting pre-dictable and financially ruinous behavior.

To see just how pervasive such behavioral biases canbe, consider a slightly modified version of the experimentpsychologists Kahneman and Tversky [1979] conducted 25years ago. Suppose you are offered two investment oppor-tunities, A and B. A yields a sure profit of $240,000, and Bis a lottery ticket yielding $1 million with a 25% probabil-

ity and $0 with 75% probability. If you had to choosebetween A and B, which would you prefer? Investment Bhas an expected value of $250,000, which is higher than A’spayoff, but this may not be all that meaningful to you becauseyou will receive either $1 million or zero. Clearly, there isno right or wrong choice here; it is simply a matter of per-sonal preferences.

Faced with this choice, most subjects prefer A, the sureprofit, to B, despite the fact that B offers a significant prob-ability of winning considerably more. This behavior is oftencharacterized as risk aversion for obvious reasons.

Now suppose you are faced with another two choices,C and D: C yields a sure loss of $750,000, and D is a lot-tery ticket yielding $0 with 25% probability and a loss of $1million with 75% probability. Which would you prefer?This situation is not as absurd as it might seem at first glance;many financial decisions involve choosing between the lesserof two evils. In this case, most subjects choose D, despite thefact that D is more risky than C. When faced with twochoices that both involve losses, individuals seem to be risk-seeking, not risk-averse as in the case of A versus B.

The fact that individuals tend to be risk-averse in theface of gains and risk-seeking in the face of losses can leadto some very poor financial decisions. To see why, observethat the combination of choices A plus D is equivalent to asingle lottery ticket yielding $240,000 with 25% probabilityand –$760,000 with 75% probability, while the combinationof choices B plus C is equivalent to a single lottery ticket yield-ing $250,000 with 25% probability and –$750,000 with 75%probability. The B plus C combination has the same proba-bilities of gains and losses, but the gain is $10,000 higher andthe loss is $10,000 lower. In other words, B plus C is formallyequivalent to A plus D plus a sure profit of $10,000. In lightof this analysis, would you still prefer A plus D?

A common response to this example is that it is con-trived, because the two pairs of investment opportunities arepresented sequentially, not simultaneously. Yet in a typicalglobal financial institution, the London office may be facedwith choices A and B and the Tokyo office may be faced withchoices C and D. Locally, it may seem as if there is no rightor wrong answer—the choice between A and B or C andD seems to be simply a matter of personal risk preferences—but the globally consolidated financial statement for theentire institution will tell a very different story.

From that perspective, there is a right answer and awrong answer, and the empirical and experimental evidencesuggests most individuals tend to select the wrong answer.Therefore, according to the behavioralists, quantitative mod-els of efficient markets—all predicated on rational choice—are likely to be wrong as well.

Grossman [1976] and Grossman and Stiglitz [1980] goeven farther. They argue that perfectly informationally effi-cient markets are an impossibility, for if markets are perfectly



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efficient, there is no profit to gathering information, inwhich case there would be little reason to trade, and mar-kets would eventually collapse.

Alternatively, the degree of market inefficiency determinesthe effort investors are willing to expend to gather and tradeon information. Hence, a non-degenerate market equilibriumwill arise only when there are sufficient profit opportunities,i.e., inefficiencies, to compensate investors for the costs of trad-ing and information gathering. The profits earned by atten-tive investors may be viewed as economic rents that accrueto those willing to engage in such activities.

Who are the providers of these rents? Black [1986] gaveus a provocative answer: “noise traders,” individuals who tradeon what they consider to be information that is, in fact,merely noise.

The supporters of the EMH have responded to thesechallenges by arguing that while behavioral biases and cor-responding inefficiencies do arise from time to time, thereis a limit to their prevalence and impact because of oppos-ing forces dedicated to exploiting such opportunities. Asimple example of such a limit is the so-called Dutch book,where irrational probability beliefs give rise to guaranteedprofits for the savvy investor.

Consider, for example, an event E, defined as “the S&P500 index drops by 5% or more next Monday,” and supposean individual has irrational beliefs as follows: There is a 50%probability that E will occur, and a 75% probability that Ewill not occur. This is clearly a violation of one of the basicaxioms of probability theory—the probabilities of two mutu-ally exclusive and exhaustive events must sum to one—butmany experimental studies have documented such violationsamong an overwhelming majority of human subjects.

These inconsistent subjective probability beliefs implythat the individual would be willing to take both of the fol-lowing bets B1and B2:

(1)

where Ec denotes the event “not E.” Now suppose we take the opposite side of both bets,

placing $50 on B1 and $25 on B2. If E occurs, we lose $50on B1 but gain $75 on B2, yielding a profit of $25. If Ec

occurs, we gain $50 on B1 and lose $25 on B2, also yieldinga profit of $25. Regardless of the outcome, we have secureda profit of $25, an arbitrage that comes at the expense of theindividual with inconsistent probability beliefs.

Such beliefs are not sustainable, and market forces—

Bif E

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namely, arbitrageurs such as hedge funds and proprietary trad-ing groups—will take advantage of these opportunities untilthey no longer exist, that is, until the odds are in line withthe axioms of probability theory.2

Therefore, proponents of the classical EMH argue thatthere are limits to the degree and persistence of behavioralbiases such as inconsistent probability beliefs, as well as sub-stantial incentives for those who can identify and exploit suchoccurrences. While all of us are subject to certain behavioralbiases from time to time, according to EMH supporters mar-ket forces will always act to bring prices back to rational lev-els, implying that the impact of irrational behavior on financialmarkets is generally negligible and, therefore, irrelevant.

But this last conclusion relies on the assumption thatmarket forces are powerful enough to overcome any type ofbehavioral bias, or, equivalently, that irrational beliefs are notso pervasive as to overwhelm the capacity of arbitrage capi-tal dedicated to taking advantage of them. This is an empir-ical issue that cannot be settled theoretically, but must be testedthrough careful measurement and statistical analysis.

One piece of anecdotal evidence is provided by the col-lapse of fixed-income relative-value hedge funds in 1998 suchas Long-Term Capital Management (LTCM). The defaultby Russia on its government debt in August 1998 triggereda global flight to quality, widening credit spreads to recordlevels and causing massive dislocation in fixed-income andcredit markets.

During that period, bonds with virtually identical cashflows and supposedly little credit risk traded at dramaticallydifferent prices, implying extraordinary profit opportunitiesto those who could afford to maintain spread positions bypurchasing the cheaper bonds and shorting the richer bonds,yielding a positive carry at the outset. If held to maturity,these spread positions would have generated payments andobligations that offset each other exactly, hence they werestructured as near-arbitrages—just like the Dutch bookexample above.

But as credit spreads widened, the gap between the longand the short side increased because illiquid bonds becamecheaper and liquid bonds became more expensive, causingbrokers and other creditors to require holders of these spreadpositions to either post additional margin or liquidate a por-tion of their positions to restore their margin levels. Thesemargin calls caused many hedge funds to start unwindingsome of their spread positions, causing spreads to widenfurther, which led to more margin calls, more unwinding,and so on. This created a cascade effect that ended with thecollapse of LTCM and several other notable hedge funds.

In retrospect, even the most ardent critics of LTCMand other fixed-income relative-value investors nowacknowledge that their spread positions were quite rational,and that their downfall was largely due to an industrywideunderappreciation of the commonality of their positions



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and the degree of leverage applied across the many hedgefunds, investment banks, and proprietary trading groupsengaged in these types of spread trades. This suggests that theforces of irrationality—investors flocking to safety and liq-uidity in the aftermath of the Russian default in August1998—were stronger, at least for several months, than theforces of rationality.

This example, and many similar anecdotes of specu-lative bubbles, panics, manias, and market crashes—a classicreference is Kindleberger [1989]—have cast reasonable doubton the hypothesis that an aggregate rationality will always beimposed by market forces.

So what does this imply for the EMH?

THE SOCIOLOGY OF MARKET EFFICIENCY

To see how a reconciliation between the EMH and itsbehavioral critics might come about, it is useful to digressto consider the potential origins of this controversy. Althoughthere are no doubt many factors contributing to this debate,one of the most compelling explanations involves key dif-ferences in the cultural and sociological aspects of economicsand psychology, which are surprisingly deep even thoughboth fields are concerned with human behavior.

Consider, first, some of the defining characteristics ofpsychology (from the perspective of an economist):

• Psychology is based primarily on observation andexperimentation.

• Field experiments are common.• Empirical analysis leads to new theories.• There are multiple theories of behavior.• Mutual consistency among theories is not critical.

Contrast these with the comparable characteristics ofeconomics:

• Economics is based primarily on theory andabstraction.

• Field experiments are not common.• Theories lead to empirical analysis.• There are few theories of behavior.• Mutual consistency is highly prized.

Although there are of course exceptions to these gen-eralizations, they do capture much of the spirit of the twodisciplines.3 For example, while psychologists certainly dopropose abstract theories of human behavior from time totime, the vast majority of academic psychologists conductexperiments. Although experimental economics has madeimportant inroads into the mainstream of economics andfinance, the top journals still publish only a small fraction ofexperimental papers; the majority is more traditional theo-

retical and empirical studies. And although new theories of economic behavior

have been proposed from time to time, most graduate pro-grams in economics and finance teach only one such the-ory: expected utility theory and rational expectations, andits corresponding extensions, e.g., portfolio optimization, thecapital asset pricing model, and dynamic general equilibriumasset pricing models. It is only recently that departures fromthis theory are not rejected out of hand. Less than a decadeago, manuscripts describing models of financial markets witharbitrage opportunities were routinely rejected at the top eco-nomics and finance journals, in some cases without even areview.

The fact that economics is still dominated by a singlemodel is a testament to the remarkable achievements of oneperson: Paul A. Samuelson. In 1947, Samuelson publishedhis Ph.D. thesis titled Foundations of Economic Analysis, whichmight have seemed somewhat arrogant were it not for thefact that it did, indeed, become the foundations of moderneconomic analysis. Much of the economic literature of thetime was based on somewhat informal discourse and dia-grammatic exposition. Samuelson, however, developed aformal mathematical framework for economic analysis thatcould be applied to a number of seemingly unrelated con-texts. His opening paragraph makes this intention explicit:

The existence of analogies between central features of vari-ous theories implies the existence of a general theory whichunderlies the particular theories and unifies them with respectto those central features. This fundamental principle ofgeneralization by abstraction was enunciated by theeminent American mathematician E.H. Moore morethan thirty years ago. It is the purpose of the pages thatfollow to work out its implications for theoretical andapplied economics [1947, p.3; italics in the original].

Samuelson then proceeded to build the infrastructureof what is now called microeconomics, which is taught asthe first graduate-level course in every Ph.D. program in eco-nomics today, and along the way, made major contributionsto welfare economics, general equilibrium theory, compar-ative static analysis, and business cycle theory.

If there is a single theme to Samuelson’s thesis, it is thesystematic application of scientific principles to economicanalysis, much like the approach of modern physics. This wasno coincidence. In Samuelson’s account of the intellectualorigins of his dissertation, he acknowledges:

Perhaps most relevant of all for the genesis of Foun-dations, Edwin Bidwell Wilson (1879–1964) was atHarvard. Wilson was the great Willard Gibbs’s last(and, essentially only) protégé at Yale. He was amathematician, a mathematical physicist, a mathe-matical statistician, a mathematical economist, a poly-



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math who had done first-class work in many fields ofthe natural and social sciences. I was perhaps his onlydisciple… I was vaccinated early to understand thateconomics and physics could share the same formalmathematical theorems (Euler’s theorem on homo-geneous functions, Weierstrass’s theorems on con-strained maxima, Jacobi determinant identitiesunderlying Le Chatelier reactions, etc.), while still notresting on the same empirical foundations and cer-tainties [1998, p. 1,376].

In a footnote to his statement regarding the generalprinciple of comparative static analysis, Samuelson adds:

It may be pointed out that this is essentially the methodof thermodynamics, which can be regarded as a purelydeductive science based upon certain postulates(notably the First and Second Laws of Thermody-namics). [1947, p. 21].

And much of the economics and finance literature sinceFoundations has followed Samuelson’s lead in attempting todeduce implications from certain postulates such as utilitymaximization, the absence of arbitrage, or the equalizationof supply and demand. In fact, the most recent milestone ineconomics—rational expectations—is founded on a singlepostulate, around which an entire literature has developed.

This cultural bias in economics, also known as “physicsenvy,” is, I claim, largely responsible for the controversybetween EMH supporters and critics. The supporters pointto the power of theoretical arguments such as expected util-ity theory, the principle of no arbitrage, and general equi-librium theory, while the latter point to experimentalevidence to the contrary.

A case in point is the Random Walk Hypothesis,which was taken to be synonymous with the EMH prior toLeroy [1973], Rubinstein [1976], and Lucas [1978], andeven several years afterward. A number of well-knownempirical studies had long since established the fact thatmarkets were weak-form efficient in Roberts’s [1967] ter-minology, implying that past prices could not be used to fore-cast future price changes.4

And although some of these studies did find evidenceagainst the random walk, e.g., Cowles and Jones [1937], theywere largely dismissed as statistical anomalies, or not eco-nomically meaningful after accounting for transaction costs,e.g., Cowles [1960]. For example, after conducting an exten-sive empirical analysis of runs of U.S. stock returns from 1956to 1962, Fama [1965, p. 80] concluded that: “there is no evi-dence of important dependence from either an investmentor a statistical point of view.”

It was in this milieu that Lo and MacKinlay [1988]reexamined the Random Walk Hypothesis, rejecting it forweekly U.S. stock returns indexes from 1962 to 1985. The

surprising element of their analysis was not only that therejections were based on fairly well-known properties ofreturns—ratios of variances of different holding periods—but also the strong reaction that their results provoked amongsome of their senior colleagues (see Lo and MacKinlay[1999, Chapter 1] for further details).

Moreover, Lo and MacKinlay [1999] observed that afterthe publication of their article they discovered several otherstudies that also rejected the Random Walk Hypothesis,and that the departures from the random walk uncovered byLarson [1960], Alexander [1961], Cootner [1962], Osborne[1962], Steiger [1964], Niederhoffer and Osborne [1966],and Schwartz and Whitcomb [1977], to name just a fewexamples, were largely ignored by the academic financecommunity.

Lo and MacKinlay provide an explanation:

With the benefit of hindsight and a more thoroughreview of the literature, we have come to the con-clusion that the apparent inconsistency between thebroad support for the Random Walk Hypothesis andour empirical findings is largely due to the commonmisconception that the Random Walk Hypothesis isequivalent to the Efficient Markets Hypothesis, andthe near religious devotion of economists to the lat-ter (see Chapter 1). Once we saw that we, and ourcolleagues, had been trained to study the data throughthe filtered lenses of classical market efficiency, itbecame clear that the problem lay not with our empir-ical analysis, but with the economic implications thatothers incorrectly attributed to our results—unbound-ed profit opportunities, irrational investors, and thelike [1999, p. 14].

The legendary trader and squash player Victor Nieder-hoffer pointed to similar forces at work in creating thisapparent cultural bias in favor of the Random Walk Hypoth-esis in an incident that took place while he was a finance PhDstudent at the University of Chicago in the 1960s (Nieder-hoffer [1997, p. 270]):

This theory and the attitude of its adherents foundclassic expression in one incident I personally observedthat deserves memorialization. A team of four of themost respected graduate students in finance had joinedforces with two professors, now considered venera-ble enough to have won or to have been consideredfor a Nobel prize, but at that time feisty as Hades andinsecure as a kid on his first date. This elite group wasstudying the possible impact of volume on stock pricemovements, a subject I had researched. As I wascoming down the steps from the library on the thirdfloor of Haskell Hall, the main business building, Icould see this Group of Six gathered together on a



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stairway landing, examining some computer output.Their voices wafted up to me, echoing off the stonewalls of the building. One of the students was point-ing to some output while querying the professors,“Well, what if we really do find something? We’ll beup the creek. It won’t be consistent with the randomwalk model.” The younger professor replied, “Don’tworry, we’ll cross that bridge in the unlikely event wecome to it.”

I could hardly believe my ears—here were six scien-tists openly hoping to find no departures from igno-rance. I couldn’t hold my tongue, and blurted out,“I sure am glad you are all keeping an open mindabout your research.” I could hardly refrain fromgrinning as I walked past them. I heard mutteredimprecations in response.

To Samuelson’s credit, he was well aware of the limi-tations of a purely deductive approach even as he wrote theFoundations, and in his introduction he offered warning:

Only the smallest fraction of economic writings, the-oretical and applied, has been concerned with thederivation of operationally meaningful theorems. In partat least this has been the result of the bad method-ological preconceptions that economic laws deducedfrom a priori assumptions possessed rigor and validityindependently of any empirical human behavior. Butonly a very few economists have gone so far as this.The majority would have been glad to enunciatemeaningful theorems if any had occurred to them. Infact, the literature abounds with false generalization.

We do not have to dig deep to find examples. Liter-ally hundreds of learned papers have been written onthe subject of utility. Take a little bad psychology, adda dash of bad philosophy and ethics, and liberal quan-tities of bad logic, and any economist can prove thatthe demand curve for a commodity is negativelyinclined [1947, p. 3, italics in the original].

This remarkable passage seems as germane today as itwas over 50 years ago when it was first written. One inter-pretation is that a purely deductive approach may not alwaysbe appropriate for economic analysis. As impressive as theachievements of modern physics are, physical systems areinherently simpler than economic systems; hence deductionbased on a few fundamental postulates is likely to be moresuccessful in the former than in the latter. Conservationlaws, symmetry, and the isotropic nature of space are pow-erful ideas in physics that simply do not have exact coun-terparts in economics.

Alternatively, imagine the impact on the explanatorypower of physical theories if relations like F = ma were tovary with the business cycle, Federal Reserve policy, or

changes in the U.S. tax code. Economic systems involvehuman interactions, which almost by definition are morecomplex than interactions of inanimate objects governed byfixed and known laws of motion. Because human behavioris heuristic, adaptive, and not completely predictable—at leastnot nearly to the same extent as physical phenomena—modeling the joint behavior of many individuals is far morechallenging than modeling just one individual. Indeed, thebehavior of even a single individual can be baffling at times,as we all know.

ADAPTIVE MARKETS: THE NEW SYNTHESIS

The sociological backdrop of the EMH debate suggeststhat an alternative to the traditional deductive approach ofneoclassic economics might be necessary. One particularlypromising direction is the application of evolutionary prin-ciples to financial markets as suggested by Farmer and Lo[1999] and Farmer [2002]. This approach is heavily influ-enced by recent advances in the emerging discipline of“evolutionary psychology,” which builds on the seminalresearch of E.O. Wilson [1975] in applying the principles ofcompetition, reproduction, and natural selection to socialinteractions, yielding surprisingly compelling explanationsfor certain kinds of human behavior such as altruism, fair-ness, kin selection, language, mate selection, religion, moral-ity, ethics, and abstract thought (see, for example, Barkow,Cosmides, and Tooby [1992]; Pinker [1993, 1997]; Craw-ford and Krebs [1998]; Buss [1999]; and Gigerenzer [2000]).

“Sociobiology” is the rubric Wilson [1975] gave tothese powerful ideas, which generated a considerable degreeof controversy in their own right, and the same principlescan be applied to economic and financial contexts. In doingso, we can fully reconcile the EMH with all its behavioralalternatives, leading to a new synthesis: the Adaptive Mar-kets Hypothesis.

Students of the history of economic thought will recallthat Thomas Malthus used biological arguments—the fact thatpopulations increase at geometric rates while natural resourcesincrease at only arithmetic rates—to arrive at rather direeconomic consequences, and that both Darwin and Wallacewere influenced by these arguments (see Hirshleifer [1977]for further details). Also, Joseph Schumpeter’s [1937] viewsof business cycles, entrepreneurs, and capitalism have anunmistakable evolutionary flavor to them; in fact, his notionsof creative destruction and bursts of entrepreneurial activityare similar in spirit to natural selection and Eldredge andGould’s [1972] notion of “punctuated equilibrium.”

More recently, economists and biologists have begunto explore these connections in several veins: direct exten-sions of sociobiology to economics (Becker [1976]; Hirsh-leifer [1977]; Tullock [1979]; evolutionary game theory(Maynard Smith [1982]; Weibull [1995]); evolutionary eco-



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nomics (Nelson and Winter [1982]; Andersen [1994];Englund [1994]; Luo [1999]); and economics as a complexsystem (Anderson, Arrow, and Pines [1988]). Hodgson[1995] provides additional examples of studies at the inter-section of economics and biology, and publications like theJournal of Evolutionary Economics and the Electronic Journal ofEvolutionary Modeling and Economic Dynamics now provide ahome for this burgeoning literature.

Evolutionary concepts have also appeared in a num-ber of financial contexts. Luo [1995, 1998, 2001, 2003]explores the implications of natural selection for futuresmarkets, and Hirshleifer and Luo [2001] consider the long-run prospects of overconfident traders in a competitive secu-rities market. The literature on agent-based modelingpioneered by Arthur et al. [1997] that simulates interactionsamong software agents programmed with simple heuristics,relies heavily on evolutionary dynamics.

And at least two prominent practitioners have proposedDarwinian alternatives to the EMH. In a chapter titled “TheEcology of Markets,” Niederhoffer [1997, Ch. 15] likensfinancial markets to an ecosystem with dealers as “herbivores,”speculators as “carnivores,” and floor traders and distressedinvestors as “decomposers.” And Bernstein [1998] makes acompelling case for active management by pointing out thatthe notion of equilibrium, which is central to the EMH, israrely realized in practice and that market dynamics are bet-ter explained by evolutionary processes.

Clearly the time is now ripe for an evolutionary alter-native to market efficiency.

To that end, we begin, as Samuelson [1947] did, withthe theory of the individual consumer. Contrary to theneoclassic postulate that individuals maximize expected util-ity and have rational expectations, an evolutionary perspec-tive makes considerably more modest claims, viewingindividuals as organisms that have been honed, through gen-erations of natural selection, to maximize the survival of theirgenetic material (see, for example, Dawkins [1976]).

While such a reductionist approach might degenerateinto useless generalities, e.g., the molecular biology of eco-nomic behavior, there are valuable insights to be gainedfrom a broader biological perspective. Specifically, this per-spective implies that behavior is not necessarily intrinsic andexogenous, but evolves by natural selection and depends onthe particular environment through which selection occurs.That is, natural selection operates not only upon geneticmaterial, but also upon social and cultural norms in Homosapiens, hence Wilson’s term, “sociobiology.”

To operationalize this perspective within an economiccontext, consider the idea of bounded rationality first espousedby Nobel Prize winning economist Herbert Simon. Simon[1955] suggests that individuals are hardly capable of the kindof optimization that neoclassic economics calls for in the stan-dard theory of consumer choice. Instead, he argues that

because optimization is costly and humans are naturally lim-ited in their computational abilities, they engage in some-thing he calls “satisficing,” an alternative to optimization inwhich individuals make choices that are merely satisfactory,not necessarily optimal. In other words, individuals arebounded in their degree of rationality, which is in sharp con-trast to the current orthodoxy—rational expectations—where individuals have unbounded rationality (the termhyperrational expectations might be more descriptive).

Unfortunately, although this idea garnered a NobelPrize for Simon, it had relatively little impact on the economicsprofession at the time. Apart from the sociological factors dis-cussed above, Simon’s framework was commonly dismissedbecause of one specific criticism: What determines the pointat which an individual stops optimizing and reaches a satis-factory solution? If such a point is determined by the usualcost/benefit calculation underlying much of microeconomics(i.e., optimize until the marginal benefits of the optimum equalthe marginal cost of getting there), this assumes the optimalsolution is known, which would eliminate the need for sat-isficing. As a result, the idea of bounded rationality fell by thewayside, and rational expectations has become the de facto stan-dard for modeling economic behavior under uncertainty.5

An evolutionary perspective provides the missing ingre-dient in Simon’s framework. The proper response to thequestion of how individuals determine the point at whichtheir optimizing behavior is satisfactory is this: Such pointsare determined not analytically, but through trial and errorand, of course, natural selection. Individuals make choicesbased on past experience and their best guess as to what mightbe optimal, and they learn by receiving positive or negativereinforcement from the outcomes. If they receive no suchreinforcement, they do not learn. In this fashion, individu-als develop heuristics to solve various economic challenges,and as long as those challenges remain stable, the heuristicswill eventually adapt to yield approximately optimal solutionsto them.

If, on the other hand, the environment changes, itshould come as no surprise that the heuristics of the old envi-ronment are not necessarily well suited to the new. In suchcases, we observe “behavioral biases”—actions that are appar-ently ill advised in the context in which we observe them.But rather than labeling such behavior irrational, we shouldrecognize that suboptimal behavior is not unlikely when wetake heuristics out of their evolutionary context. A moreaccurate term for such behavior might be “maladaptive.” Theflopping of a fish on dry land may seem strange and unpro-ductive, but underwater, the same motions are capable of pro-pelling the fish away from its predators.

By coupling Simon’s notion of bounded rationality andsatisficing with evolutionary dynamics, many other aspectsof economic behavior can also be derived. Competition,cooperation, market-making behavior, general equilibrium,



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and disequilibrium dynamics are all adaptations designed toaddress certain environmental challenges for the humanspecies, and by viewing them through the lens of evolutionarybiology, we can better understand the apparent contradic-tions between the EMH and the presence and persistenceof behavioral biases.

Specifically, the Adaptive Markets Hypothesis can beviewed as a new version of the EMH, derived from evolu-tionary principles. Prices reflect as much information asdictated by the combination of environmental conditions andthe number and nature of “species” in the economy or, touse a more appropriate biological term, the ecology.

By species, I mean distinct groups of market participants,each behaving in a common manner. For example, pensionfunds may be considered one species; retail investors another;market makers a third; and hedge fund managers a fourth.

If multiple species (or the members of a single highlypopulous species) are competing for rather scarce resourceswithin a single market, that market is likely to be highly effi-cient, e.g., the market for 10-year U.S. Treasury notes,where most relevant information is incorporated into priceswithin minutes. If, on the other hand, a small number ofspecies are competing for rather abundant resources in a givenmarket, that market will be less efficient, e.g., the market foroil paintings from the Italian Renaissance. Market efficiencycannot be evaluated in a vacuum, but is highly context-dependent and dynamic, just as insect populations advanceand decline as a function of the seasons, the number ofpredators and prey they face, and their abilities to adapt toan ever-changing environment.

The profit opportunities in any given market are akinto the amount of food and water in a particular localecology—the more resources present, the less fierce thecompetition. As competition increases, either because ofdwindling food supplies or an increase in the animal popu-lation, resources are depleted, which in turn causes a pop-ulation decline, eventually reducing the level of competitionand starting the cycle again. In some cases cycles convergeto corner solutions; i.e., certain species become extinct,food sources are permanently exhausted, or environmentalconditions shift dramatically. By viewing economic profitsas the ultimate food source on which market participantsdepend for their survival, the dynamics of market interac-tions and financial innovation can be readily derived.

Under the AMH, behavioral biases abound. The ori-gins of such biases are heuristics that are adapted to non-financial contexts, and their impact is determined by the sizeof the population with such biases versus the size of com-peting populations with more effective heuristics.

During the fall of 1998, the desire for liquidity andsafety by a certain population of investors overwhelmed thepopulation of hedge funds attempting to arbitrage such pref-erences, causing those arbitrage relations to break down. In

the years prior to August 1998, however, fixed-income rel-ative-value traders profited handsomely from these activities,presumably at the expense of individuals with seemingly irra-tional preferences (in fact, such preferences were shaped bya certain set of evolutionary forces, and might have been quiterational in other contexts).

Therefore, under the AMH, investment strategiesundergo cycles of profitability and loss in response to chang-ing business conditions, the number of competitors enter-ing and exiting the industry, and the type and magnitude ofprofit opportunities available. As opportunities shift, so toowill the affected populations. For example, after 1998, thenumber of fixed-income relative-value hedge funds declineddramatically—because of outright failures, investor redemp-tions, and fewer startups in this sector—but many have reap-peared in recent years as the performance of this type ofinvestment strategy has improved.

Even fear and greed—the two most common culpritsin the downfall of rational thinking, according to mostbehavioralists—are the product of evolutionary forces,adaptive traits that enhance the probability of survival.Recent research in the cognitive neurosciences and eco-nomics suggests an important link between rationality indecision-making and emotion (Grossberg and Gutowski[1987]; Damasio [1994]; Elster [1998]; Lo [1999]; Loewen-stein [2000]; Peters and Slovic [2000]; and Lo and Repin[2002]), implying that the two are not antithetical, but infact complementary.

For example, contrary to the common belief thatemotions have no place in rational financial decision-mak-ing processes, Lo and Repin [2002] present preliminary evi-dence that physiological variables associated with theautonomic nervous system are highly correlated with mar-ket events even for highly experienced professional securi-ties traders. They argue that emotional responses are asignificant factor in the real-time processing of financialrisks, and that an important component of a professionaltrader’s skills lies in his or her ability to channel emotion, con-sciously or unconsciously, in specific ways during certain mar-ket conditions.

This argument often surprises economists because ofthe ostensible link between emotion and behavioral biases,but a more sophisticated view of the role of emotions inhuman cognition shows that they are central to rationality(see, for example, Damasio [1994] and Rolls [1990, 1994,1999]). In particular, emotions are the basis for a reward andpunishment system that facilitates the selection of advanta-geous behavior, providing a numeraire for animals to engagein a “cost-benefit analysis” of the various actions open tothem (Rolls [1999, Chapter 10.3]). From an evolutionaryperspective, emotion is a powerful adaptation that dramati-cally improves how efficiently animals learn from their envi-ronments and their pasts.6



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These evolutionary underpinnings are more than sim-ple speculation in the context of financial market participants.The extraordinary degree of competitiveness of global finan-cial markets and the outsize rewards that accrue to the“fittest” traders suggest that Darwinian selection—“survivalof the richest,” to be precise—is at work in determining thetypical profile of the successful trader. After all, unsuccess-ful traders are eventually eliminated from the population aftersuffering a certain level of losses.

The AMH is still under development, and certainlyrequires much more research to render it “operationally mean-ingful” in Samuelson’s sense. Even at this early stage, though,it seems clear that an evolutionary framework is able to reconcilemany of the apparent contradictions between efficient marketsand behavioral exceptions. The former may be viewed as thesteady-state limit of a population with constant environmen-tal conditions, and the latter involves specific adaptations of cer-tain groups that may or may not persist, depending on theparticular evolutionary paths that the economy experiences.

More specific implications may be derived through acombination of deductive and inductive inferences—forexample, theoretical analysis of evolutionary dynamics, empir-ical analysis of evolutionary forces in financial markets, andexperimental analysis of decision-making at the individual andgroup level—and are currently under investigation.

PRACTICAL IMPLICATIONS

Despite the rather abstract and qualitative nature of theAMH presented above, a number of surprisingly concreteimplications can be derived.

The first implication is that, to the extent that there isa relation between risk and reward, it is unlikely to be sta-ble over time. Such a relation is determined by the relativesizes and preferences of various populations in the marketecology, as well as institutional aspects such as the regulatoryenvironment and tax laws. As these factors shift over time,any risk/reward relation is likely to be affected.

A corollary of this implication is that the equity riskpremium is also time-varying and path-dependent. This isnot so revolutionary an idea as it might first appear—evenin the context of a rational expectations equilibrium model,if risk preferences change over time, then the equity risk pre-mium must vary too.

The incremental insight of the AMH is that aggregaterisk preferences are not universal constants, but are shapedby the forces of natural selection. For example, until recently,U.S. markets were populated by a significant group ofinvestors who have never experienced a genuine bear mar-ket—this fact has undoubtedly shaped the aggregate risk pref-erences of the U.S. economy, just as the experience of thelast four years, since the bursting of the technology bubblehas affected the risk preferences of the current population

of investors. In this context, natural selection determines whoparticipates in market interactions; those investors who expe-rienced substantial losses in the technology bubble are morelikely to have exited the market, leaving a different popula-tion of investors today than four years ago.

Through the forces of natural selection, history mat-ters. Irrespective of whether prices fully reflect all availableinformation, the particular path that market prices havetaken over the past few years influences current aggregate riskpreferences.

Among the three Ps of Total Investment Manage-ment, preferences is clearly the most fundamental and leastunderstood. Several large bodies of research have developedaround these issues—in economics and finance, psychology,operations research (also called decision sciences) and, morerecently, brain and cognitive sciences—and many new insightsare likely to flow from synthesizing these different strands ofresearch into a more complete understanding of how indi-viduals make decisions. Simon’s [1982] seminal contributionsto this literature are still remarkably timely and their impli-cations have yet to be fully explored.7

A second implication is that, contrary to the classicalEMH, arbitrage opportunities do arise from time to time inthe AMH. As Grossman and Stiglitz [1980] observe, with-out such opportunities, there will be no incentive to gatherinformation, and the price discovery aspect of financial mar-kets will collapse.

From an evolutionary perspective, the very existenceof active liquid financial markets implies that profit oppor-tunities must be present. As they are exploited, they disap-pear. But new opportunities are also constantly being createdas certain species die out, as others are born, and as institu-tions and business conditions change.

Rather than the inexorable trend toward higher effi-ciency predicted by the EMH, the AMH implies consider-ably more complex market dynamics, with cycles as well astrends, and panics, manias, bubbles, crashes, and other phe-nomena that are routinely witnessed in natural market ecolo-gies. These dynamics provide the motivation for activemanagement as Bernstein [1998] suggests, and give rise toNiederhoffer’s [1997] “carnivores” and “decomposers.”

A third implication is that investment strategies will alsowax and wane, performing well in certain environments andperforming poorly in other environments. Contrary to theclassical EMH in which arbitrage opportunities are competedaway, eventually eliminating the profitability of the strategydesigned to exploit the arbitrage, the AMH implies thatsuch strategies may decline for a time, and then return toprofitability when environmental conditions become moreconducive to such trades.

An obvious example is risk arbitrage, which has beenunprofitable for several years because of the decline in invest-ment banking activity since 2001. As M&A activity begins



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to pick up again, however, risk arbitrage will start to regainits popularity among both investors and portfolio managers,as it has just this year.

A more striking example can be found by comput-ing the rolling first-order autocorrelation r1 of monthlyreturns of the S&P composite index from January 1871through April 2003 (see the Exhibit). As a measure of mar-ket efficiency, r1 might be expected to take on larger valuesduring the early part of the sample and become progressivelysmaller during recent years as the U.S. equity marketbecomes more efficient. (Recall that the random walkhypothesis implies that returns are serially uncorrelated,hence r1 should be 0 in theory).

It is apparent from the Exhibit, however, that thedegree of efficiency—as measured by the first-order auto-correlation—varies through time in a cyclical fashion, andthere are periods in the 1950s when the market is more effi-cient than in the early 1990s.

Such cycles are not ruled out by the EMH in theory,but in practice none of its empirical implementations hasincorporated these dynamics, assuming instead that theworld is stationary and markets are perpetually in equilib-rium. This widening gulf between the stationary EMH andobvious shifts in market conditions no doubt contributed toBernstein’s [2003] recent critique of the policy portfolio instrategic asset allocation models and his controversial proposalto reconsider the case for tactical asset allocation.

A fourth implication is that innovation is the key tosurvival. The classic EMH suggests that certain levels ofexpected returns can be achieved simply by bearing a suffi-cient degree of risk. The AMH implies that because therisk/reward relation varies through time, a better way toachieve a consistent level of expected returns is to adapt tochanging market conditions. By evolving a multiplicity ofcapabilities that are suited to a variety of environmentalconditions, investment managers are less likely to becomeextinct as a result of rapid changes in business conditions.Consider the current theory of the demise of the dinosaurs,and ask where the next financial killer asteroid might comefrom (see Alvarez [1997]).

Finally, the AMH has a clear implication for all finan-cial market participants. Survival is the only objective thatmatters. While profit maximization, utility maximization, andgeneral equilibrium are certainly relevant aspects of marketecology, the organizing principle in determining the evolu-tion of markets and financial technology is simply survival.

There are many other practical insights and potentialbreakthroughs that can be derived from the AMH as we shiftour mode of thinking in financial economics from the phys-ical to the biological sciences. Although evolutionary ideasare not yet part of the financial mainstream, my hope is thatthey will become more commonplace as they demonstratetheir worth—ideas are also subject to survival of the fittest.

No one has illustrated this principle so well as Harry


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E X H I B I TRolling 60-Month First-Order Autocorrelation Coefficient rr11 of Monthly S&P Composite Returns—January 1987–April 2003

Data source: Robert J. Shiller, available at http://www.econ.yale.edu/~shiller/.


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Markowitz, the father of modern portfolio theory and aNobel laureate in economics in 1990. In describing hisexperience as a Ph.D. student on the eve of his graduation,he wrote in his Nobel address:

When I defended my dissertation as a student in theEconomics Department of the University of Chicago,Professor Milton Friedman argued that portfolio the-ory was not Economics, and that they could notaward me a Ph.D. degree in Economics for a disser-tation which was not Economics. I assume that he wasonly half serious, since they did award me the degreewithout long debate. As to the merits of his arguments,at this point I am quite willing to concede: at the timeI defended my dissertation, portfolio theory was notpart of Economics. But now it is [1991, p. 476].

Perhaps over the next 30 years, The Journal of Portfo-lio Management will also bear witness to the relevance of theAdaptive Markets Hypothesis for financial markets andeconomics.

ENDNOTES

The author thanks Stephanie Hogue, Dmitry Repin, andSvetlana Sussman for helpful comments and discussion. Researchsupport from the MIT Laboratory for Financial Engineering isgratefully acknowledged.

1Parts of this article include ideas and exposition from mypublished research. Where appropriate, I have modified passagesto suit the current context without detailed citations and quota-tion marks so as to preserve continuity. Readers interested in theoriginal sources may consult Lo [1997, 1999, 2002], Lo andMacKinlay [1999], and Lo and Repin [2002].

2Only when these axioms are satisfied is arbitrage ruled out.This was conjectured by Ramsey [1926] and proved rigorously byde Finetti [1937] and Savage [1954].

3For a less impressionistic and more detailed comparison ofpsychology and economics, see Rabin [1998, 2002].

4See, for example, Kendall [1953], Osborne [1959, 1962],Roberts [1959, 1967], Cowles [1960], Larson [1960], Working[1960], Alexander [1961, 1964], Granger and Morgenstern [1963],Mandelbrot [1963], Fama [1965], Fama and Blume [1966], andCowles and Jones [1937].

5Simon’s work is now receiving greater attention, thanks inpart to the growing behavioral literature in economics and finance.See, for example, Simon [1982], Sargent [1993], Rubinstein[1998], Gigerenzer et al. [1999], Gigerenzer and Selten [2001], andEarl [2002].

6This important insight was forcefully illustrated by Dama-sio [1994] in his description of one of his patients, code-named Elliot,who underwent surgery to remove a brain tumor. Along with thetumor, part of his frontal lobe had to be removed as well, and afterhe recovered from the surgery, it was discovered that Elliot no longerpossessed the ability to experience emotions of any kind. Thisabsence of emotional response had a surprisingly profound effect

on his day-to-day activities, as Damasio [1994, p. 36] describes:

When the job called for interrupting an activity andturning to another, he might persist nonetheless,seemingly losing sight of his main goal. Or he mightinterrupt the activity he had engaged, to turn tosomething he found more captivating at that partic-ular moment…. The flow of work was stopped. Onemight say that the particular step of the task at whichElliot balked was actually being carried out too well,and at the expense of the overall purpose. One mightsay that Elliot had become irrational concerning thelarger frame of behavior.

Apparently, Elliot’s inability to feel—his lack of emotionalresponse—rendered him irrational from society’s perspective.

7 More recent research on preferences include Kahneman,Slovic, and Tversky [1982], Hogarth and Reder [1986], Gigeren-zer and Murray [1987], Dawes [1988], Fishburn [1988], Keeneyand Raiffa [1993], Plous [1993], Sargent [1993], Thaler [1993],Damasio [1994], Arrow et al. [1996], Laibson [1997], Picard[1997], Pinker [1997], and Rubinstein [1998]. Starmer [2000]provides an excellent review of this literature.

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Globally Evolutionarily Stable Portfolio Rules∗

Igor V. Evstigneev a, Thorsten Hens b andKlaus Reiner Schenk-Hoppe c

AbstractThe paper examines a dynamic model of a financial market with en-dogenous asset prices determined by short run equilibrium of supplyand demand. Assets pay dividends that are partially consumed andpartially reinvested. The traders use fixed-mix investment strategies(portfolio rules), distributing their wealth between assets in fixed pro-portions. Our main goal is to identify globally evolutionarily stablestrategies, allowing an investor to “survive,” i.e. to accumulate in thelong run a positive share of market wealth, regardless of the initialstate of the market. It is shown that there is a unique portfolio rulewith this property—an analogue of the famous Kelly (1956) rule of“betting one’s beliefs.” A game theoretic interpretation of this resultis given.

JEL-Classification: G11, C61, C62.Keywords: Evolutionary Finance, Wealth Dynamics, Survival and Extinctionof Portfolio Rules, Evolutionary Stability, Kelly Rule.

∗Financial support by the national center of competence in research “Financial Valua-tion and Risk Management” is gratefully acknowledged. The national centers in researchare managed by the Swiss National Science Foundation on behalf of the federal authorities.

aEconomics, School of Social Sciences, University of Manchester, Oxford Road,Manchester M13 9PL, United Kingdom. E-mail: [email protected].

bSwiss Banking Institute, University of Zurich, Plattenstrasse 32, CH-8032 Zurich,Switzerland. E-mail: [email protected].

cLeeds University Business School and School of Mathematics, University of Leeds,Leeds LS2 9JT, United Kingdom. E-mail: [email protected].

1

1 Introduction

1.1. Evolutionary finance. Price changes and dividend payments of stocksinduce wealth dynamics among investors using different investment strate-gies (portfolio rules) in financial markets. These dynamics act as a naturalselection force among the portfolio rules: some prove to be successful and“survive,” the others fail and “become extinct.” The purpose of the presentpaper is to investigate financial dynamics from this evolutionary perspec-tive with the view to identifying evolutionarily stable (surviving) investmentstrategies.

Evolutionary ideas have a long history in the social sciences going backto Malthus, who played an inspirational role for Darwin (for a review of thesubject see, e.g., Hodgeson [20]). A more recent stage of development of theseideas began in the 1950s with the publications of Alchian [2], Penrose [32]and others. A powerful momentum to work in this area was given by the in-terdisciplinary research conducted in the 1980s and 1990s under the auspicesof the Santa Fe Institute in New Mexico, USA, where researchers of differ-ent backgrounds—economists, mathematicians, physicists and biologists—combined their efforts to study evolutionary dynamics in biology, economicsand finance; see, e.g., Arthur, Holland, LeBaron, Palmer and Taylor [5],Farmer and Lo [16], LeBaron, Arthur and Palmer [24], Blume and Easley [7],and Blume and Durlauf [6].

Questions of survival and extinction of portfolio rules have been studiedby Blume and Easley [7, 8] and Sandroni [35] in general equilibrium modelswith perfect foresight (e.g. Laffont [23], Ch. 6), where agents maximize dis-counted sums of expected utilities. The selection results in their papers aredriven by the interplay between an agent’s consumption and the accuracyof his subjective beliefs (i.e. the individual assessment of the probabilities offuture states). Most of the positive results in that line of research pertain tothe case where the markets under consideration are complete.

The approach to evolutionary finance pursued here marks a departurefrom the conventional general equilibrium paradigm. In the model we dealwith, the asset market dynamics are determined by the dynamic interactionof strategies of the traders, rather than by the maximization of utilities ofconsumption. These strategies are taken as fundamental characteristics ofthe agents, while the optimality of individual behavior and the coordinationof beliefs (or the lack of it) are not reflected in formal terms but are ratherleft to the interpretation of the observed behavior. A specific feature ofthis approach is that it rests only on model components that are observableand can be estimated empirically, which makes the theory closer to practicalapplications. Our modeling framework based on random dynamical systems

2

is substantially new, but one can trace its connections to some classical ideasin economics (we revive in the new context the Marshallian [28] concept oftemporary equilibrium—see the discussion in Section 2). Although one doesnot require notions of rationality to define the dynamics, the main result hasa game theoretic interpretation (Theorem 2), which links this study to thetheory of market games—Shapley and Shubik [37] and others.

1.2. Outline of the model and the results. The model we proposedescribes the dynamics of a financial market in which there are I investors(traders) and K traded assets (securities). Asset supply is constant over time.Each trader chooses a strategy prescribing to distribute, at the beginning ofeach time period t = 1, 2, ..., his/her investment budget between the securi-ties according to given proportions. Assets pay dividends, that are randomand depend on a discrete-time stochastic process of exogenous “states of theworld.”

The prices of the securities at each date are derived endogenously fromthe equilibrium condition: aggregate market demand of each asset is equal toits supply. Each investor’s individual demand depends on his/her budget andinvestment proportions. The main results pertain to the case where theseproportions are fixed (constant over time). The budget of each investordepends on time and random factors. It has two sources: the dividends paidby the assets and capital gains. These two sources form investor’s wealth,which is partially consumed and partially reinvested at each time period.When analyzing the long-run performance of trading strategies, we assumethat the investment/consumption ratio is fixed and that it is the same for allthe traders.

We note that the class of fixed-mix, or constant proportions, strategieswe consider in this work is quite common in financial theory and practice;see, e.g., Perold and Sharpe [33], Mulvey and Ziemba [29], Browne [10] andDempster [12, 13]. From the theoretical standpoint, this class of strategiesprovides a convenient laboratory for the analysis of questions we are inter-ested in. It makes it possible to formalize in a clear and compact way theconcept of the type of an investor which determines the evolutionary per-formance of his/her portfolio rule in the long run. A similar approach iscommon in evolutionary game theory (e.g. Weibull [40]), and in this paperwe initiate the analysis of our model by pursuing it in the context of an assetmarket dynamics.

The strategy profile of the investors determines the “ecology” of the mar-ket and its random dynamics over time. In the evolutionary perspective,survival or extinction of investment strategies is governed by the long-runbehavior of the relative wealth of the investors, which depends on the com-bination of the strategies chosen. A portfolio rule (or an investor using it)

3

is said to survive if it accumulates in the limit a positive fraction of totalmarket wealth. It is said to become extinct if the share of market wealthcorresponding to it tends to zero.

An investment strategy, or a portfolio rule, is called evolutionarily stableif the following condition holds. If a group of investors uses this rule, whileall the others use different ones, those and only those investors survive whobelong to the former group. If this condition holds regardless of the initialstate of the market, the investment strategy is called globally evolutionarilystable. If it holds under the additional assumption that the group of investorsusing other portfolio rules (distinct from the one we consider) possesses asufficiently small initial share of market wealth, then the above property ofstability is termed local.

We prove that among all fixed-mix (i.e. constant proportions) investmentstrategies, the only globally evolutionarily stable portfolio rule is to investaccording to the proportions of the expected dividends. This recipe is similarto the well-known Kelly’s principle of “betting one’s beliefs.” The presentpaper contributes to that field of studies which originated from the pioneer-ing work of Shannon1 and Kelly [22]—see Breiman [9], Thorp [39], Algoetand Cover [3], Hakansson and Ziemba [18] and references therein. Most ofthe previous work was concerned with models where asset prices were givenexogenously, or where the analysis was based on a reduction to such models[7]. Our aim is to obtain analogous results in a dynamic equilibrium setting,with endogenous prices. Intermediate steps towards this aim were made inthe previous papers [4] and [14]. Those papers dealt with a special case of“short-lived” assets. Here, we extend the results to a model with long-lived,dividend-paying assets and thus achieve the long-sought goal of providing anatural and general framework for this class of results.

1.3. Plan of the paper. The structure of the paper is as follows.Section 2 provides a rigorous description of the model, a brief outline ofwhich was given above. In Section 3, we formulate and discuss the mainresults. Section 4 develops methods needed for the analysis of the modelunder consideration. The Appendix contains proofs of the technical resultsstated in Section 4.

1Although Claude Shannon—the famous founder of the mathematical theory ofinformation—did not publish on investment-related issues, his ideas, expressed in his lec-tures on investment problems, should apparently be regarded as the initial source of thatstrand of literature which we cite here. For the history of these ideas and the relateddiscussion see Cover [11].

4

2 The model

2.1. Equilibrium dynamics of an asset market. In the model we dealwith, there are I ≥ 2 investors (traders) acting in a market where K ≥ 2risky assets are traded. The situation on the market at date t = 0, 1, ... ischaracterized by the set of vectors

(pt; x1t , ..., x

It )

where pt ∈ RK+ is the vector of market prices of the assets and xi

t =(xi

t,1, ..., xit,K) ∈ RK

+ is the portfolio of investor i. For each k = 1, ..., K,the coordinate pt,k of the vector pt = (pt,1, ..., pt,K) stands for the price of oneunit of asset k at date t. The coordinate xi

t,k of the vector xit = (xi

t,1, ..., xit,K)

indicates the amount of (“physical units” of) asset k in the portfolio xit. The

scalar product 〈pt, xit〉 =

∑Ki=1 pt,kx

it,k expresses the market value of investor

i’s portfolio at date t.Investor i’s behavior (and implicitly his/her preferences) at date t are

characterized by the demand function X it(pt, x

it−1), assigning to each pair

of vectors pt ∈ RK+ and xi

t−1 ∈ RK+ the vector xi

t = X it(pt, x

it−1) ∈ RK

+ . Ifthe investor possessed the portfolio xi

t−1 at date t − 1 (“yesterday”), thenhe/she will be willing to purchase the portfolio xi

t = X it(pt, x

it−1) at date t

(“today”), provided that today’s asset price system is pt. All the coordinatesof X i

t(pt, xit−1) are non-negative: borrowing and short sales are ruled out.

Define the aggregate demand function for the market under considerationas

Xt(pt, xt−1) :=I∑

i=1

X it(pt, x

it−1), (1)

where xt−1 = (x1t−1, ..., x

It−1) is the set of portfolios of all the investors. It

is supposed that the supply of each asset in each time period is constantand, for simplicity, normalized to 1. We examine the equilibrium marketdynamics, assuming that, in each time period, the demand on each asset isequal to its supply:

I∑i=1

X it,k(pt, x

it−1) = 1, k = 1, ..., K.

By using the notation (1) and e := (1, 1, ..., 1), the previous system of equa-tions can be written in the vector form:

Xt(pt, xt−1) = e. (2)

5

Each solution to this system, pt, is an equilibrium price vector. It will followfrom the assumptions we are going to impose that this vector exists, is uniqueand is strictly positive for each xt−1 satisfying

xt−1 = (x1t−1, ..., x

It−1) ∈ RK×I

+ ,

I∑i=1

xit−1 = e. (3)

Equations (2) and

xit = X i

t(pt, xit−1), i = 1, ..., I, (4)

define a dynamical system describing the evolution of the asset market. Thestate of this dynamical system at time t = 1, 2, ... is a pair (pt, xt), wherext = (x1

t , ..., xIt ) ∈ RK×I

+ is a collection of investors’ portfolios satisfying∑Ii=1 xi

t = e and pt is a non-negative K-dimensional vector whose coordinatesare the equilibrium asset prices prevailing at date t. For date 0, the set ofinitial endowments (initial portfolios) x0 = (x1

0, ..., xI0) satisfying (3) is given.

The knowledge of the state xt−1 at time t − 1 allows us to compute theequilibrium price vector pt at time t as the solution to equation (2). Basedon pt and xt−1, we can determine xt = (x1

t , ..., xIt ) from (4). By iterating this

procedure, we can generate a path

(p1, x1), (p2, x2), ... (5)

of the dynamical system in question.In the model considered here, the demand functions X i

t(·, ·) of investorsdepend on random factors (via random asset dividends). Therefore paths (5)are random processes, and so we will deal with a random dynamical system.We are interested, in particular, in the long-run behavior (as t →∞) of pathsof this system in connection with questions of evolutionary market dynamics.

2.2. Investors’ budgets and demand functions. In the model un-der consideration, investor i’s budget at date t, on which his/her demanddepends, is given by

Bit(pt, x

it−1) := 〈dt, x

it−1〉+ 〈pt, x

it−1〉. (6)

There are two sources from which the budget is formed: the value 〈pt, xit−1〉

of yesterday’s portfolio xit−1, expressed in terms of the current prices pt, and

the dividend

〈dt, xit−1〉 =

K∑k=1

dt,kxit−1,k

6

yielded by the portfolio xit−1. It is supposed that (one unit of) asset k pays

the dividend dt,k ≥ 0 at date t ≥ 1. The dividends depend on random factorsas described below.

Randomness is modeled as follows. There is a finite set S and a sequences0, s1, ... of random variables with values in S. The random parameter st

characterizes the state of the world at date t. The dividends dt,k of the assetsk = 1, ..., K are supposed to be functions of the history st := (s0, ..., st) ofstates of the world prior to date t:

dt,k = dt,k(st) ≥ 0 (k = 1, ..., K, t = 1, 2, ...).

We impose the following fundamental assumptions regarding dt,k(·):(d1) for all st, we have

∑Kk=1 dt,k(s

t) > 0;(d2) for each k = 1, 2, ..., K, the expectation Edt,k(s

t) is strictly positive.

The first assumption means that always at least one asset pays a strictlypositive dividend. According to the second condition, for each asset k theprobability that it pays a strictly positive dividend is strictly positive.

We assume that the individual demand function of investor i is of theform

X it,k(pt, x

it−1) = µi

t,k

Bit(pt, x

it−1)

pt,k

, (7)

where µit,1, ..., µ

it,K are nonnegative numbers satisfying

µit,1 + ... + µi

t,K < 1. (8)

According to (6) and (7), the trader acts in the market as follows. He/she getsthe dividends 〈dt, x

it−1〉 from the portfolio xi

t−1 and rebalances the portfolio(by selling some assets and buying others) at the prices pt so that the newportfolio xi

t = X it(pt, x

it−1) satisfies

pkt X

it,k(pt, x

it−1) = µi

t,kBit(pt, x

it−1).

Thus the portfolio xit is constructed by investing the fraction µi

t,k of the budgetBi

t(pt, xit−1) (given by (6))) into the kth position xi

t,k of xit.

If pt,k = 0, then the expression on the right-hand side of (7) is not defined,and in this case we put X i

t,k(pt, xit−1) = 0. We define the price pt,k of asset k

as zero if this asset is not traded; in this case, the holdings of this asset inthe portfolio of each investor are zero, i.e. X i

t,k(pt, xit−1) = 0. This possibility,

however, will be excluded by the assumptions we are going to impose. Underthese assumptions all the equilibrium asset prices will be strictly positive.

7

According to (8), the sum µit,1 + ... + µi

t,K is strictly less than one. Thismeans that not all of the investor’s budget is used for purchasing assets.Some fraction of the budget is used for consumption. This fraction,

µit,0 := 1− (µi

t,1 + ... + µit,K), (9)

is the investor i’s consumption rate.2.3. Investors’ strategies and the random dynamical system

they generate. The vectors µit = (µi

t,1, ..., µit,K) describing trader i’s invest-

ment decisions can depend on the observed states of the world: µit = µi

t(st).

A trading (investment) strategy of investor i is a sequence of non-zero non-negative vector functions

µit(s

t) = (µit,1(s

t), ..., µit,K(st)), t = 1, 2, ...

satisfying (8) for all t and all realizations of the states of the world. Aslong as the vectors dt = (dt,1, ..., dt,K), t = 1, 2, ..., depend on st, the de-mand functions (7) also depend on st, and so the state (pt, xt) of the randomdynamical system under consideration is a function of the history st of thestates of the world from time zero to time t. If a strategy profile (µt(s

t))∞t=1 =(µ1

t (st), ..., µI

t (st))∞t=1 of all the investors is fixed, the random sate of the mar-

ket (xt(st), pt(s

t)) (where xt(st) = (x1

t (st), ..., xI

t (st))) evolves according to

the following system of equations

pt,k =I∑

i=1

µit,k〈dt + pt, x

it−1〉, k = 1, 2, ..., K; (10)

pt,kxit,k = µi

t,k〈dt + pt, xit−1〉. (11)

The vector pt = pt(st) ∈ RK

+ satisfying (10) exists and is unique. Thisfollows from the fact that, for each st and each vector xi

t−1 satisfying (3),the operator transforming a vector p = (p1, ..., pK) ∈ RK

+ into the vectorq = (q1, ..., qK) ∈ RK

+ with coordinates

qk =I∑

i=1

µit,k〈dt + p, xi

t−1〉 (12)

is contracting in the norm |p| :=∑

k |pk|. Indeed,

|q − q′| =K∑

k=1

|qk − q′k| =K∑

k=1

|I∑

i=1

µit,k〈p− p′, xi

t−1〉| ≤ (maxi

K∑k=1

µit,k)|p− p′|,

(13)

8

where maxi

∑Kk=1 µi

t,k < 1 (see (8)) and |〈p − p′, xit−1〉| ≤ |p − p′| because

0 ≤ xit−1,k ≤ 1 (see (3)).

The unique price vector pt ≥ 0 satisfying (10) is strictly positive if one ofthe following conditions holds:

(I) µit,k > 0, 0 ≤ xi

0 6= 0;

(II)∑I

i=1 µik > 0, dt,k(s

t) > 0, 0 ≤ xi0 6= 0.

Indeed, if (I) is valid, then we have

pt,k =I∑

i=1

µit,k〈dt + pt, x

it−1〉 ≥ min

j(µj

t,k)I∑

i=1

〈dt + pt, xit−1〉 ≥

minj

(µjt,k)

I∑i=1

〈dt, xit−1〉 = min

j(µj

t,k)〈dt, e〉 > 0

because 〈dt, e〉 =∑K

k=1 dt,k > 0 by virtue of (d1). Once pt > 0, the budget ofeach investor 〈dt + pt, x

it−1〉 is strictly positive, provided that 0 ≤ xi

t−1 6= 0.Hence, xi

t > 0 for each t ≥ 1. (All inequalities between vectors are understoodcoordinatewise.)

Suppose (II) holds. Then we have

pt,k =I∑

i=1

µit,k〈dt + pt, x

it−1〉 ≥ (

I∑i=1

µit,k) min

i〈dt + pt, x

it−1〉.

If 0 ≤ xit−1 6= 0, then 〈dt, x

it−1〉 > 0, and so pt > 0. Furthermore, xi

t,k 6= 0 for

some k because xit,k = (pt,k)

−1∑I

i=1 µit,k〈dt + pt, x

it−1〉 and µi

t,k > 0 for somek.

The validity of at least one of conditions (I) or (II) is essentially necessaryfor the dynamical system in question to be non-degenerate in the followingsense. If neither condition is supposed to hold, then it might happen thatpt,k = 0 for some t and k. Indeed, denote by ei the vector whose coordinatesare equal to 0 except the ith coordinate which is equal to 1 and put K = I =2, xi

0 = ei, d1 = e1 and µi1 = ei/2. Then p1,2 = µ1

1,2〈d1, x10〉 + µ2

1,2〈d1, x20〉 =

0 ·1+(1/2) ·0 = 0. Throughout the paper, we are going to deal with a modelin which condition (I) holds; condition (II) is provided here only for the sakeof completeness.

2.4. Market evolution and Marshallian temporary equilibrium.In the model we deal with, the dynamics of an asset market is modeled interms of a sequence of temporary equilibria. At each date t the investors’strategies µi

t,k, the asset dividends dt,k and the portfolios xit−1 determine—in

accordance with (10)—the asset prices pt = (p1t , ..., p

Kt ) equilibrating aggre-

gate asset demand and supply. The asset holdings xit−1 = (xi

t−1,1, ..., xit−1,K)

9

play the role of initial endowments available at the beginning of date t. Theportfolios xi

t constructed according to (11) are transferred to date t + 1 andthen in turn serve as initial endowments for the investors.

The dynamics of the asset market described above are similar to thedynamics of the commodity market as outlined in the classical treatise byAlfred Marshall [28] (Book V, Chapter II “Temporary Equilibrium of De-mand and Supply”). Marshall’s ideas were introduced into formal economicsby Samuelson [34], pp. 321–323. Equations analogous to (10), (11) werederived in continuous time by Lotka [26]; they became the classics of math-ematical evolutionary biology. In rigorous terms, the Marshallian concept oftemporary equilibrium, as it is understood in the present work, was in muchdetail analyzed in an economics context by Schlicht [36]. The equations onpp. 29–30 in the monograph [36] are direct continuous-time, deterministiccounterparts of our equations (10) and (11). They underlie the Marshalliantemporary equilibrium approach.

As it was noticed by Samuelson [34] and emphasized by Schlicht [36],in order to study the process of market dynamics by using the Marshallian“moving equilibrium method,” one needs to distinguish between at leasttwo sets of economic variables changing with different speeds. Then theset of variables changing slower (in our case, the set xt = (x1

t , ...., xIt ) of

investors’ portfolios) can be temporarily fixed, while the other (in our case,the asset prices pt) can be assumed to rapidly reach the unique state of partialequilibrium. Samuelson [34] writes about this approach:

I, myself, find it convenient to visualize equilibrium processes ofquite different speed, some very slow compared to others. Within eachlong run there is a shorter run, and within each shorter run there isa still shorter run, and so forth in an infinite regression. For analyticpurposes it is often convenient to treat slow processes as data andconcentrate upon the processes of interest. For example, in a shortrun study of the level of investment, income, and employment, it isoften convenient to assume that the stock of capital is perfectly orsensibly fixed.

As it follows from the above citation, Samuelson thinks about a hierarchyof various equilibrium processes with different speeds. In our model, it issufficient to deal with only two levels of such a hierarchy. We leave theprice adjustment process leading to the solution of the partial equilibriumproblem (10) beyond the scope of the model. It can be shown, however,that this equilibrium will be reached at an exponential rate in the courseof any naturally defined tatonnement procedure (cf. (12) and (13)). Results

10

yielding a rigorous justification of the above approach, involving “rapid” and“slow” variables, are provided in continuous time by the theory of singularperturbations—e.g. Smith [38].

The concept of a temporary, or moving, equilibrium was introduced andanalyzed apparently for the first time by Marshall. However, in the last fourdecades this term has been by and large associated with a different notion,going back to Lindahl [25] and Hicks [19]. This notion was developed informal settings by Grandmont and others (see, e.g., the volume [17]). Thecharacteristic feature of the Lindahl-Hicks temporary equilibrium is the ideaof forecasts or beliefs about the future states of the world, which all themarket participants possess and which are formalized in terms of stochastickernels (transition functions) conditioning the distributions of future states ofthe world upon the agents’ private information. A discussion of the modernstate of this direction of research is provided by Magill and Quinzii [27].

In this work, we pursue a completely different approach. Our modelmight indirectly take into account agents’ forecasts or beliefs, but they canbe only implicitly incorporated into the agents’ investment strategies. Thesestrategies are the only agents’ characteristics we rely upon in our modeling.Such characteristics can be observed and estimated based on the real marketdata, and we formulate our results in the form of recommendations for aninvestor what strategies to follow.

To distinguish the above approach to temporary equilibrium from the onebased on the Hicks–Lindahl concepts, we suggest the terms “Marshallian”or “evolutionary temporary equilibrium.” The former term is motivated bythe already cited Marshall’s [28] ideas. The latter is justified not only by themain focus of the model on questions of survival and extinction of portfoliorules, but also by deep analogies between the dynamic processes governedby the evolutionary equations (10), (11) and similarly described processesin evolving complex systems in physics, mechanics, chemical kinetics, evolu-tionary biology, ecology and other sciences—see, e.g., Hofbauer and Sigmund[21].

3 The main results

3.1. Dynamics of market shares. Our further analysis will be based onthe following assumptions:

(i) the states of the world s0, s1, s2, ... form a sequence of independentidentically distributed (i.i.d.) elements in S such that the probability P{st =s} is strictly positive for each s ∈ S;

(ii) the asset dividends dt,k(st) are functions of the current state st of the

11

world:dt,k(s

t) = Dk(st),

where the functions Dk(s) ( s ∈ S, k = 1, 2, ..., K) do not depend on t, arenon-negative and satisfy

(D1)∑K

k=1 Dk(s) > 0, s ∈ S;(D2) EDk(st) > 0, k = 1, ..., K.

Each investor i chooses an investment strategy (portfolio rule) char-acterized by a fixed non-negative vector µi = (µi

1, ..., µiK) such that 0 <

µi1+...+µi

K < 1. The numbers µik indicate the fractions of investor i’s budget

according to which he/she distributes wealth between the assets k = 1, ..., K.These fractions remain the same over time, so that we deal here with simple,or fixed-mix, investment strategies. In the remainder of the paper we willconsider only those portfolio rules (µi

1, ..., µiK) which are completely mixed,

i.e., µik > 0 for each k = 1, ...K. Furthermore, we will assume that the

consumption rate µi0 = 1 −

∑Kk=1 µi

k is the same for all the investors andit is equal to 1 − ρ, where ρ is some given number in (0, 1). We supposethat the consumption rate is the same for all the market traders because weare mainly interested in comparing the long-run performance of investmentstrategies. This can be done only for a group of traders having the sameconsumption rate. Otherwise, a seemingly lower performance of a strategymay be simply due to a higher consumption rate of the investor.

As long as the sum∑K

k=1 µik does not depend on i and is equal to some

given number ρ ∈ (0, 1), it is convenient to characterise the investment deci-sions of each investor i in terms of the vector of investment proportions

λi = (λi1, ..., λ

iK), λi

k := µik/ρ.

The numbers λik are strictly positive and λi

1 + ...+λiK = 1. The set of vectors

whose coordinates satisfy these conditions will be denoted by ∆K+ . From now

on, we will associate the terms “investment strategy” or “portfolio rule” withsuch vectors of investment proportions.

If each investor i = 1, ..., I selects a portfolio rule λi = (λi1, ..., λ

iK) ∈ ∆K

+ ,the strategy profile (λ1, ..., λI) determines, in accordance with equations

pt,k = ρI∑

i=1

λik〈dt + pt, x

it−1〉, k = 1, 2, ..., K, (14)

pt,kxit,k = ρλi

k〈dt + pt, xit−1〉, k = 1, 2, ..., K, (15)

equivalent to (10) and (11), the random path of market dynamics (pt, xt),t = 1, 2, .... Both the prices pt = (pt,1, ..., pt,K) and the asset holdings xi

t =

12

(xit,1, ..., x

it,K) depend on st—the history of the states of the world prior to

time t.Denote by

wit := 〈pt + dt, x

it−1〉 (16)

investor i’s wealth available for consumption and investment at date t ≥ 1.The total market wealth is equal to Wt =

∑Ii=1 wi

t. (From formulas (18)and (19) below, it follows that Wt > 0.) We are primarily interested inthe long-run behavior of the relative wealth, or market shares, ri

t := wit/Wt

of the traders, i.e. in the asymptotic properties of the sequence of vectorsrt = (r1

t , ..., rIt ) as t → ∞. To analyze these properties, we will derive

equations allowing to compute the vector rt+1 based on the knowledge of thevector rt and the state of the world st+1 realized at date t + 1.

From (14) we get

pt+1,k = ρI∑

i=1

λik〈dt+1 + pt+1, x

it〉 = ρ

I∑i=1

λikw

it+1 = ρ〈λk, wt+1〉,

xit,k =

λikw

it

〈λk, wt〉, k = 1, 2, ..., K,

where λk := (λ1k, ..., λ

Ik) and wt := (w1

t , ..., wIt ). Consequently,

wit+1 :=

K∑k=1

[pt+1,k + Dk(st+1)]xit =

K∑k=1

[ρ〈λk, wt+1〉+ Dk(st+1)]λi

kwit

〈λk, wt〉. (17)

By summing up these equations over i = 1, ..., I, we obtain

Wt+1 =K∑

k=1

[ρ〈λk, wt+1〉+ Dk(st+1)]

∑Ii=1 λi

kwit

〈λk, wt〉= ρWt+1 +

K∑k=1

Dk(st+1).

This leads to the formula

Wt+1 =D(st+1)

1− ρ, (18)

where

D(st+1) =K∑

k=1

Dk(st+1) (> 0) (19)

13

is the sum of the dividends of all the assets. Dividing both sides of equation(17) by Wt+1 and using formula (18), we find

rit+1 =

K∑k=1

[ρ〈λk, rt+1〉+ (1− ρ)Dk(st+1)

D(st+1)]

λikw

it/Wt

〈λk, wt〉/Wt

.

Thus we arrive at the system of equations:

rit+1 =

K∑k=1

[ρ〈λk, rt+1〉+ (1− ρ)Rk(st+1)]λi

krit

〈λk, rt〉, i = 1, ..., I, (20)

where

Rk(st+1) =Dk(st+1)

D(st+1), k = 1, ..., K,

are the relative dividends of the assets k = 1, ..., K.Let ∆I denote the set of vectors r = (r1, ..., rI) ≥ 0 whose norm |r| :=∑|ri| is equal to one. It can be shown (see Section 4 below) that, for any

rt = (r1t , ..., r

It ) ∈ ∆I and any st+1 ∈ S, the system of equations (20) has

a unique solution rt+1 ≥ 0. We have rt+1 ∈ ∆I , which can be verified bysumming up equations (20) and using the fact that

∑Kk=1 Rk(s) = 1, s ∈ S.

We will denote the solution rt+1 to system (20) (as a function of st+1 andrt) by F (st+1, rt). The mapping F (st+1, ·) transforms ∆I into ∆I . Thus wedeal here with a random dynamical system

rt+1 = F (st+1, rt) (21)

on the unit simplex ∆I . We will assume that a strictly positive non-randomvector r0 ∈ ∆I is fixed. Starting from this initial state, we can generate apath (trajectory) r0, r1(s

1), r2(s2), ... of the random system (21) according to

the formulart+1(s

t+1) = F (st+1, rt(st)), t = 0, 1, ...,

(If t = 0, we formally write r0 = r0(s0) having in mind that r0 is a constant.)

Remark. The model for the dynamics of investors’ market shares wehave described above was proposed in [15]. Its presentation in this paper isslightly different from that in [15]. (In particular, we here write ρ in placeof 1 − λ0 and λi

k instead of λik/(1 − λ0) in [15].) In the limit as ρ → 0, the

model reduces to the one studied in [14]. In particular, if ρ = 0, the randomdynamical system described by equations (20) coincides with that examinedin [14].

3.2. Survival and extinction of portfolio rules. We examine thedynamics of the relative wealth ri

t, governed by equations (20), from an

14

evolutionary perspective. We are interested in questions of “survival andextinction” of portfolio rules. We say that a portfolio rule λi = (λi

1, ..., λiK)

(or investor i using it) survives with probability one in the market selectionprocess (20) if, for the relative wealth ri

t of investor i, we have limt→∞ rit >

0 almost surely. We say that λi becomes extinct with probability one iflimt→∞ ri

t = 0 almost surely.A central role in this work is played by the following definition.

Definition 1 A portfolio rule λ = (λ1, ..., λK) is called globally evolution-arily stable if the following condition holds. Suppose, in a group of investorsi = 1, 2, ..., J (1 ≤ J < I), all use the portfolio rule λ, while all the others,i = J + 1, ..., I use portfolio rules λi distinct from λ. Then those investorswho belong to the former group (i = 1, ..., J) survive with probability one,whereas those who belong to the latter (i = J + 1, ..., I) become extinct withprobability one.

In the above definition, it is supposed that the initial state r0 in themarket selection process governed by equations (20) is any strictly positivevector r0 ∈ ∆I . This is reflected in the term “global evolutionary stability.”An analogous local concept (cf. [15]) is defined similarly, but in the definitionof local evolutionary stability, the initial market share rJ+1

0 + ... + rI0 of the

group of investors who use strategies λi distinct from λ is supposed to besmall enough.

Our main goal is to identify that portfolio rule which is globally evolu-tionarily stable. Clearly, if it exists it must be unique. Indeed if there aretwo such rules, λ 6= λ′, we can divide the population of investors into twogroups assuming that the first uses λ and the second λ′. Then, accordingto the definition of global evolutionary stability, both groups must becomeextinct with probability one, which is impossible since the sum of the relativewealth of all the investors is equal to one.

3.3. Central result. Define

λ∗ = (λ∗1, ..., λ∗K), λ∗k = ERk(st), k = 1, ..., K,

so that λ∗1, ..., λ∗K are the expected relative dividends of assets k = 1, ..., K.

The portfolio rule (investment strategy) λ∗ is called the Kelly portfolio rule.It prescribes to invest in accordance with the principle of “betting one’sbeliefs,” as formulated in the pioneering paper by Kelly [22], for furtherstudies in this direction see Breiman [9], Thorp [39], Algoet and Cover [3]and Hakansson and Ziemba [18].

Recall that, according to a convention made in Section 2, we considerin this paper only completely mixed portfolio rules. Therefore the vectors

15

λ and λi involved in Definition 1 are supposed to be strictly positive. TheKelly rule is completely mixed by virtue of assumptions (D1) and (D2).

Throughout the paper, we will assume that the functions R1(s), ..., RK(s)are linearly independent (there are no redundant assets).

The main result of this paper is as follows.

Theorem 1 The Kelly rule is globally evolutionarily stable.

In order to prove this theorem we have to consider a group of investorsi = 1, ..., J using the portfolio rule λ∗, assume that all the other investors i =J+1, ..., I use portfolio rules λi 6= λ∗ and show that the former group survives,while the latter becomes extinct. In general, J should be any number between1 ≤ J < I. We note, however, that it is sufficient to prove the theoremassuming that J = 1, in which case the result reduces to the assertion thatr1t → 1 almost surely. To perform the reduction of the case J > 1 to the

case J = 1, we “aggregate” the group of investors i = 1, 2, ..., J into one bysetting

r1t = r1

t + ... + rJt .

By adding up equations (20) over i = 1, ..., J , we obtain:

r1t+1 =

K∑k=1

[ρ〈λk, rt+1〉+ (1− ρ)Rk(st+1)]λ∗kr

1t

〈λk, rt〉,

where

〈λk, r〉 = λ∗kr1 +

I∑i=J+1

λikr

i.

Thus the original model reduces to the analogous one in which there areI − J + 1 investors (i = 1, J + 1, ..., I) so that investor 1 uses the Kellystrategy λ∗ and all the others, i = J + 1, ..., I, use strategies distinct fromλ∗. If we have proved Theorem 1 in the special case J = 1, we know thatrit → 0 almost surely for all i = J + 1, ..., I. Consequently, r1

t → 1, whichmeans that the group of investors i = 1, ..., I (which we treat as a single,“aggregate,” investor) accumulates in the limit all market wealth. It remainsto observe that in the original model, the proportions between the relativewealth of investors i, j who belong to the group 1, ..., J using the Kelly ruledo not change in time. This is so because for all such investors, the growthrates of their relative wealth are the same:

rit+1

rit

=K∑

k=1

[ρ〈λk, rt+1〉+ (1− ρ)Rk(st+1)]λ∗k

〈λk, rt〉, i = 1, ..., J.

16

Consequently,rit+1

rit

=rjt+1

rjt

, i, j = 1, ..., J,

and sorit+1

r1t+1

=rit

r1t

=ri0

r10

, i = 1, ..., J.

Therefore rit = βir1

t (i = 1, ..., J) for all t, where βi = ri0/r

10 is a strictly

positive constant. Since

r1t =

J∑i=1

rit = (

J∑i=1

βi)r1t → 1 (a.s.),

we obtain that rit → βi(

∑Ji=1 βi)−1 > 0 (a.s.) for all i = 1, ..., J , which means

that all the “Kelly investors” i = 1, ..., J survive.Thus in order to prove Theorem 1 it is sufficient to establish the following

fact: if investor 1 uses the Kelly rule, while all the others use strategiesdistinct from the Kelly rule, investor 1 is almost surely the single survivor inthe market selection process. We will prove this assertion in Section 4 basedon a number of auxiliary results. These results provide methods needed forthe analysis of the model at hand, and some of them are of independentinterest.

3.4. Random asset market game. We would like to discuss ourmain result from a game-theoretic point of view. Consider a non-cooperativegame with I players (investors), whose strategies are completely mixed simpleportfolio rules λi ∈ ∆K

+ . A strategy profile Λ := (λ1, ..., λI) defines accordingto (20) a random dynamical system generating for each i the random processri0, r

i1, r

i2, ... of the market shares of trader i = 1, ..., I. Define the random

payoff function of investor i as Φi(Λ) := lim supt→∞ rit. The random variable

Φi(Λ) is always well-defined and takes values in [0,1]. These data define anon-cooperative game with random payoffs, which we will call the randomasset market game. Consider the strategy profile Λ∗ := (λ∗1, ..., λ∗I) for whichλ∗i := λ∗ (the Kelly strategy profile).

Theorem 2 The Kelly strategy profile forms with probability one a symmet-ric dominant strategy Nash equilibrium in the random asset market game.

The assertion of the theorem means that if one of the players i, say playeri = 1, employs the strategy λ∗ and all the other players use any strategiesλ2, ..., λI ∈ ∆K

+ then

Φ1(λ∗, λ2, ..., λI) ≥ Φ1(λ

1, λ2, ..., λI)

17

for any λ1 ∈ ∆K+ . In other words, investor 1 cannot be better off by deviating

from λ∗, regardless of what the other investors’ strategies are.Theorem 2 is an immediate consequence of Theorem 1. Indeed, if at least

one of investors i = 2, ..., I uses λ∗, then any deviation of investor 1 from λ∗

will imply Φ1(λ1, λ2, ..., λI) = 0, since in the limit all the market wealth will

be gathered almost surely by those who use λ∗. If only player 1 employs λ∗,then Φ1(λ

∗, λ2, ..., λI) = 1, which is not less than Φ1(λ1, λ2, ..., λI) because

this function takes values in [0, 1].A static (one-period) version of the above game in which the payoff func-

tions were defined in terms of expected payoffs was considered in the paperby Alos-Ferrer and Ania [1]. That paper dealt with a model involving shortlived assets (the case ρ = 0). The definition of the game in [1] does notinvolve the asymptotic performance of investment strategies in the long run,and in this connection, the results in [1] are different from ours.

4 Techniques for the analysis of evolutionary

market dynamics

4.1. The mapping defining the random dynamical system. In thissection we develop methods needed for the study of the model under consid-eration. We here provide only the statements and discussions of the results;their proofs are relegated to the Appendix. We begin with the analysis ofthe mapping defining the random dynamical system (20).

Let ρ be a number satisfying 0 ≤ ρ < 1. For each s ∈ S, consider themapping F (s, r) = (F 1(s, r), ..., F I(s, r)) of the unit simplex ∆I into itselfdefined by

F i(s, r) =K∑

k=1

[ρ〈λk, F (s, r)〉+ (1− ρ)Rk(s)]λi

kri

〈λk, r〉, i = 1, ..., I. (22)

The fact that the mapping under consideration is well-defined is establishedin Proposition 1 below. Fix some element s of the state space S and a vectorr = (r1, ..., rI) ∈ ∆I . Consider the affine operator B : RI

+ → RI+ transforming

a vector x = (x1, ..., xI) ∈ RI+ into the vector y = (y1, ..., yI) ∈ RI

+ defined by

yi =K∑

k=1

[ρ〈λk, x〉+ (1− ρ)Rk]λi

kri

〈λk, r〉,

where Rk = Rk(s).

18

Proposition 1 The operator B possesses a unique fixed point in RI+. This

point belongs to the unit simplex ∆I .

Proposition 2 We have

I∑i=1

|F i(s, r)〉−F i(s, r)| ≤ 1

(1− ρ)

I∑i=1

K∑k=1

| λikr

i

〈λk, r〉− λi

kri

〈λk, r〉|, r, r ∈ ∆I . (23)

It follows from (23) and the inequalities

〈λk, r〉 =I∑

i=1

λikr

i > 0, 〈λk, r〉 =I∑

i=1

λikr

i > 0,

(holding because λik > 0) that the mapping F (s, r) is continuous in r ∈ ∆I .

For each s ∈ S and r = (r1, ..., rI) ∈ ∆I , define

gi(s, r) =K∑

k=1

[ρ〈λk, F (s, r)〉+ (1− ρ)Rk(s)]λi

k

〈λk, r〉, i = 1, ..., I. (24)

It follows from (22) that if ri > 0, then

gi(s, r) =F i(s, r)

ri

so that gi(s, r) is the growth rate of the ith coordinate under the mappingF . Define

µ∗ = mini,k

λik, µ∗ = max

i,kλi

k, H = µ∗/µ∗.

The proposition below shows that the growth rate is uniformly bounded awayfrom zero and infinity.

Proposition 3 For each r ∈ ∆I and each i = 1, ..., I, we have

H−1 ≤ gi(s, r) ≤ H, s ∈ S. (25)

The function gi(s, r) is continuous in r ∈ ∆I .

4.2. Return on the Kelly portfolio. Define

f(s, r) =K∑

k=1

[ρ〈λk, F (s, r)〉+ (1− ρ)Rk(s)]λ∗k

〈λk, r〉, (26)

19

where λ∗k = ERk(s), k = 1, ..., K. Suppose, at date t, the relative wealth ofthe investors i = 1, ..., I are given by the vector r = (r1, ..., rI) ∈ ∆I . Thenthe (relative) asset prices at dates t and t + 1 are

pk = 〈λk, r〉, qk(s) = 〈λk, F (s, r)〉, (27)

provided the state of the world realized at date t + 1 is s. An investor’sportfolio in which unit wealth is distributed between the assets according tothe proportions λ∗k, k = 1, ..., K, is called the Kelly portfolio. The (gross)return on this portfolio, taking into account the dividends and consumption,is given by the function f(s, r) defined by (26), which can be written as

f(s, r) =K∑

k=1

[ρqk(s) + (1− ρ)Rk(s)]λ∗kpk

.

If one of the investors i = 1, ..., I, say investor 1, employs the investmentstrategy λ∗ = (λ∗1, ..., λ

∗K) (i.e., λ1

k = λ∗k, k = 1, ..., K), then the growth rateof his/her market share is equal to f(s, r):

g1(s, r) = f(s, r)

(see (24) and (26)).An important result on which the analysis of the model at hand is based

is contained in the following theorem.

Theorem 3 For each r ∈ ∆I , we have

E ln f(s, r) ≥ 0. (28)

This inequality is strict if and only if

〈λk, r〉 6= λ∗k for at least one k = 1, ..., K. (29)

This result means that the expected logarithmic return on the Kellyportfolio (λ∗1, ..., λ

∗K) is non-negative. It is strictly positive if and only if

(λ∗1, ..., λ∗K) does not coincide with the market portfolio (p1, ..., pK). Recall

that the total amount of each asset is normalized to 1, so that the total wealthinvested into asset k is pk = 〈λk, r〉. We emphasize that, in Theorem 3, itis not assumed that any of the investors i = 1, ..., I uses the Kelly strategy.The result is applicable without this assumption.

4.3. The Kelly portfolio and the market portfolio. According toTheorem 3, the expected logarithmic return on the Kelly portfolio (λ∗1, ..., λ

∗K)

is non-negative. It is strictly positive if and only if the market portfolio

20

(p1, ..., pK) does not coincide with (λ∗1, ..., λ∗K). Of course it can happen at

some moment of time that (λ∗1, ..., λ∗K) = (p1, ..., pK). But can it happen that

the market portfolio coincides with (λ∗1, ..., λ∗K) at two consecutive moments

of time? In other words, can the system of equalities

qk(s) = pk = λ∗k (k = 1, ..., K, s ∈ S) (30)

hold? Recall that we denote by pk the price of the asset k corresponding tothe vector r = (r1, ..., rI) of relative wealth at some fixed moment of time,

pk = 〈λk, r〉 =I∑

i=1

λikr

i

and by qk(s) the price of the asset at the next moment of time, when thestate of the world realized is s:

qk(s) = 〈λk, F (s, r)〉 =I∑

i=1

λikF

i(s, r).

The question we formulated is important for the analysis of the asymptoticbehavior of the relative wealth of an investor using the Kelly rule. As Propo-sition 4 below shows, the answer to this question (under the assumptions weimpose) is negative.

Recall that we assume that there are no redundant assets, i.e., the func-tions R1(s), ..., RK(s) are linearly independent. This assumption will be usedin the following proposition.

Proposition 4 Suppose one of the following assumptions is fulfilled.(a) All the investors i = 1, 2, ..., I use portfolio rules λi = (λi

1, ..., λiK)

distinct from the Kelly rule λ∗ = (λ∗1, ..., λ∗K).

(b) All the investors i = 2, 3, ..., I use portfolio rules λi = (λi1, ..., λ

iK) dis-

tinct from the Kelly rule λ∗ = (λ∗1, ..., λ∗K), and the wealth share r1of investor

1 is less than one.Then equations (30) cannot hold.

4.4. Limiting behavior of the Kelly investor’s relative wealth.Let r0 be a strictly positive vector in ∆I . Define recursively the sequence ofrandom vectors r0, r1(s

1), r2(s2), ... by the formula rt = F (st, rt−1). Then

rt = (r1t , ..., r

It ) is the vector of relative wealths of the investors i = 1, ..., I

at date t, depending on the realization st = (s1, ..., st) of states of the world.It follows from Proposition 3 that rt > 0 as long as rt−1 > 0 and so all the

21

vectors rt(st) are strictly positive for all t and st. Consequently, the random

variablesln ri

t = ln rit(s

t), i = 1, ..., I, t = 0, 1, ...

are well-defined and finite. Clearly, they have finite expectations becauseeach of them takes on a finite number of values (since the set S is finite).

Suppose investor 1 uses the Kelly rule

λ∗ = (λ∗1, ..., λ∗K) = (ER1(s), ..., ERK(s)).

Consider the growth rate r1t+1/r

1t of investor 1’s relative wealth. It can be

expressed as follows:

r1t+1

r1t

= g1(st+1, rt) =F 1(st+1, rt)

r1t

[rt = rt(st)]

(see (24)), and since the strategy λ1 of investor 1 coincides with the Kellyrule λ∗, we have

r1t+1

r1t

= g1(st+1, rt) = f(st+1, rt), (31)

where f(s, r) is the function defined by (26).Denote by ξt = ξt(s

t) the logarithm of the relative wealth of investor 1,ξt = ln r1

t . We claim that the sequence ξt is a submartingale:

E(ξt+1|st) ≥ ξt. (32)

Indeed, we have

E(ξt+1|st)− ξt = E[(ξt+1 − ξt)|st] = E[(ln r1t+1 − ln r1

t )|st] = E[(lnr1t+1

r1t

)|st]

= E[ln f(st+1, rt)|st] = E[ln f(s, rt)]|rt=rt(st) =∑s∈S

π(s) ln f(s, rt(st)),

where π(s) > 0 is the probability that st+1 = s. The last two equalities inthe above chain of relations follow from the fact that the random variabless1, s2, ... are independent and identically distributed. By virtue of Theorem 3,∑

s∈S π(s) ln f(s, rt(st)) ≥ 0, which proves (32). Since 0 < r1

t ≤ 1, we haveξt ≤ 0, and so ξt, t = 0, 1, ..., is a non-positive submartingale. As is well-known, a non-positive submartingale converges almost surely (a.s.) ξt → ξ∞(a.s.) as t →∞ (see, e.g., [31], Section IV.5). This implies r1

t = eξt → eξ∞ >0 (a.s.). This leads to the following result.

Theorem 4 The relative wealth of a Kelly investor converges a.s., and thelimit is strictly positive.

22

It follows from Theorem 4 that an investor using the Kelly strategy sur-vives with probability one. A key result of this study is Theorem 5 below,asserting that if one of the investors uses the Kelly rule and all the others useother strategies, distinct from the Kelly one, then the Kelly investor is theonly survivor in the market selection process. As has been noticed in 3.3,this result immediately implies Theorem 1.

Theorem 5 Let the strategy of investor 1 coincide with the Kelly rule: λ1k =

λ∗k, k = 1, ..., K. Let the strategies of investors i = 2, ..., I be distinct fromthe Kelly rule:

(λi1, ..., λ

iK) 6= (λ∗1, ..., λ

∗K).

Then the relative wealth r1t of investor 1 converges to one almost surely.

We note that if ρ = 0, Theorem 5 follows from the main result of thepaper [14]. Methods developed in this work are different in some respectsfrom those in [14]. Although they are applicable to a substantially morecomplex model, they do not give exponential estimates for the convergenceof the relative wealth process.

Appendix

The Appendix contains proofs of the results presented in the previous section.Proof of Proposition 1. Consider any x, x ∈ RI

+ and put y = B(x), y = B(x). Wehave

|y − y| =I∑

i=1

|yi − yi| = ρ

I∑i=1

|K∑

k=1

〈λk, x− x〉λi

kri

〈λk, r〉|

≤ ρK∑

k=1

I∑i=1

|〈λk, x− x〉|λi

kri

〈λk, r〉= ρ

K∑k=1

|〈λk, x− x〉|

≤ ρK∑

k=1

I∑j=1

λjk|x

j − xj | = ρI∑

j=1

|xj − xj | = ρ|x− x|.

Thus the operator B : RI+ → RI

+ is contracting and hence it contains a uniquefixed point x ∈ RI

+. To show that x ∈ ∆I we sum up the equations

xi =K∑

k=1

[ρ〈λk, x〉+ (1− ρ)Rk]λi

kri

〈λk, r〉

over i = 1, ..., I and obtain

|x| =K∑

k=1

[ρ〈λk, x〉+ (1− ρ)Rk]∑I

i=1 λikr

i

〈λk, r〉=

K∑k=1

[ρ〈λk, x〉+ (1− ρ)Rk]

23

The function gi(s, r) is continuous in r ∈ ∆I , because F (s, r) is continuous in rand 〈λk, r〉 ≥ µ∗ > 0 (see (24)). �

Proof of Theorem 3. 1st step. Multiplying both sides of (22) by λim and

summing up over i = 1, ..., I, we get

〈λm, F (s, r)〉 =K∑

k=1

[ρ〈λk, F (s, r)〉+ (1− ρ)Rk(s)]∑I

i=1 λikλ

imri

〈λk, r〉(33)

(m = 1, ...,K). By using the notation introduced in (27), equations (33) andinequality (28) can be written as

qm(s) =K∑

k=1

[ρqk(s) + (1− ρ)Rk(s)]∑I

i=1 λimλi

kri

pk, m = 1, ...,K, (34)

and

E lnK∑

k=1

[ρqk(s) + (1− ρ)Rk(s)]λ∗kpk

≥ 0. (35)

Condition (29) is necessary for this inequality to be strict (the “only if” part in(29)) because pk = λ∗k for all k = 1, ...,K implies that the left-hand side of (35) iszero.

2nd step. We fix the argument s and omit it in the notation. Consider theK ×K matrix

A = (amk), amk = δmk − ρ

∑Ii=1 λi

mλikr

i

pk,

where δmk = 1 if m = k and δmk = 0 if m 6= k. Put

b = (b1, ..., bK), bm = (1− ρ)K∑

k=1

Rk

∑Ii=1 λi

mλikr

i

pk. (36)

Then (for the fixed s) the system of equations (34) can be written

Aq = b (37)

[q = q(s)]. Indeed, the mth coordinate (Aq − b)m of the vector Aq − b can beexpressed as follows

(Aq − b)m =K∑

k=1

amkqk − bm

= qm − ρK∑

k=1

qk

∑Ii=1 λi

mλikr

i

pk− (1− ρ)

K∑k=1

Rk

∑Ii=1 λi

mλikr

i

pk

= qm −K∑

k=1

[ρqk + (1− ρ)Rk]∑I

i=1 λimλi

kri

pk,

25

which is equal to the difference between the left-hand side and the right-hand sideof (34).

We can represent the matrix A as A = Id−ρC, where Id is the identity matrixand

C = (cmk), cmk =∑I

i=1 λimλi

kri

pk.

The norm of the linear operator C is not greater than one, because

|Cx| =K∑

m=1

|K∑

k=1

cmkxk| ≤K∑

m=1

K∑k=1

cmk|xk| = |x|, (38)

whereK∑

m=1

cmk =K∑

m=1

∑Ii=1 λi

mλikr

i

pk=

∑Ii=1 λi

kri

pk= 1. (39)

Consequently, the operator ρC is contracting, and so each of the equivalent equa-tions Aq = 0 and q = ρCq has a unique solution. Thus the matrix A is non-degenerate, and the solution to the linear system (37) can be represented asq = A−1b.

3rd step. Define

ck = ρλ∗kpk

, d = (1− ρ)K∑

k=1

Rkλ∗kpk

. (40)

Then we have

〈c, q〉+ d =K∑

k=1

[ρqk + (1− ρ)Rk]λ∗kpk

. (41)

This expression appears in (35), and our goal is to estimate the expected logarithmof it (when Rk and q depend on s). To this end we write

〈c, q〉 = 〈c, A−1b〉 = 〈(A−1)′c, b〉 = 〈(A′)−1c, b〉, (42)

where A′ denotes the conjugate matrix. In (42), we use the identity (A−1)′ =(A′)−1, holding for each invertible linear operator A.

By virtue of (42),〈c, q〉 = 〈b, l〉, (43)

where l = (A′)−1c, i.e., the vector l is the solution to the linear system A′l = c.The matrix A′ is given by

A′ = (a′km), a′km = amk = δmk − ρ

∑Ii=1 λi

mλikr

i

pk,

and the linear system A′l = c can be written

K∑m=1

(δmk − ρ

∑Ii=1 λi

mλikr

i

pk)lm = ρ

λ∗kpk

, k = 1, ...,K,

26

(see (40)) or equivalently,

lk = ρ(K∑

m=1

∑Ii=1 λi

mλikr

i

pklm +

λ∗kpk

), k = 1, ...,K. (44)

Further, in view of (40) and (36), we obtain

d + 〈l, b〉 = (1− ρ)K∑

k=1

Rkλ∗kpk

+ (1− ρ)K∑

m=1

lm

K∑k=1

Rk

∑Ii=1 λi

mλikr

i

pk

= (1− ρ)K∑

k=1

Rkλ∗kpk

+ (1− ρ)K∑

k=1

Rk

K∑m=1

lm

∑Ii=1 λi

mλikr

i

pk

= (1− ρ)K∑

k=1

Rk[K∑

m=1

lm

∑Ii=1 λi

mλikr

i

pk+

λ∗kpk

] =(1− ρ)

ρ

K∑k=1

Rklk, (45)

where the last equality follows from (44). Consequently,

〈c, q〉+ d = 〈l, b〉+ d =(1− ρ)

ρ

K∑k=1

Rklk (46)

(see (41) and (43)).4th step. According to Step 1 of the proof, we have to establish inequality (35)

for every solution q(s), s ∈ S, of system (34) and show that this inequality is strictif

(p1, ..., pK) 6= (λ∗1, ..., λ∗K). (47)

The considerations presented in Steps 2 and 3, allow to reduce this problem to thefollowing one: for the solution l = (l1, ..., lK) to system (44), show that

E ln[(1− ρ)

ρ

K∑k=1

Rk(s)lk] ≥ 0

(see (46) and (43)). Additionally, it has to be shown that the last inequality isstrict if assumption (47) holds. The advantage of the new problem comparative tothe original one lies in the fact that system (44), in contrast with (34), does notdepend on s.

We write (44) equivalently as

(1− ρ)ρ

pklk = ρ[K∑

m=1

pm(1− ρ)

ρlm

I∑i=1

λimλi

kri

pm+

(1− ρ)ρ

λ∗k],

and, by changing variables

fk =(1− ρ)

ρlkpk,

27

we transform (44) to

fk = ρK∑

m=1

fm

∑Ii=1 λi

mλikr

i

pm+ (1− ρ)λ∗k, k = 1, ...,K. (48)

Then(1− ρ)

ρ

K∑k=1

Rklk =K∑

k=1

Rkfk

pk,

and the problem reduces to the following one: given the solution (f1, ..., fK) tosystem (48), show that

E lnK∑

k=1

Rk(s)fk

pk≥ 0 (49)

with strict inequality if (p1, ..., pK) 6= (λ∗1, ..., λ∗K).

Note that the affine operator defined by the right-hand side of (48) is contract-ing (see (38) and (39)) and leaves the non-negative cone RK

+ invariant. Thereforethere exists a unique vector f = (f1, ..., fK) solving (48). Furthermore, this vectoris strictly positive (which follows from the strict positivity of λ∗ = (λ∗1, ..., λ

∗K))

and satisfies∑K

k=1 fk = 1. The last equality can be obtained by summing upequations (48) over k = 1, ...,K.

By virtue of Jensen’s inequality, applied to the concave function ln(·), we have

E lnK∑

k=1

Rk(s)fk

pk≥ E

K∑k=1

Rk(s) lnfk

pk=

K∑k=1

λ∗k lnfk

pk.

(We use here the fact that∑K

k=1 Rk(s) = 1 for all s.) Thus it is sufficient to provethat if a vector (f1, ..., fK) satisfies (48), then

K∑k=1

λ∗k lnfk

pk≥ 0, (50)

and inequality (50) is strict when assumption (47) is fulfilled. This problem ispurely deterministic: no random parameter s is involved either in (48) or in (50).

5th step. Put gk = fk/pk, k = 1, ...,K. Then, from (48), we get

pkgk = ρ

K∑m=1

gm

I∑i=1

λimλi

kri + (1− ρ)λ∗k, k = 1, ...,K. (51)

Let us multiply both sides of these equations by ln gk and sum up over k = 1, ...,K:

K∑k=1

pkgk ln gk = ρ

K∑k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri + (1− ρ)

K∑k=1

λ∗k ln gk.

28

This yields

K∑k=1

λ∗k ln gk =1

(1− ρ)[

K∑k=1

pkgk ln gk − ρK∑

k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri].

Further, we haveK∑

k=1

λ∗k lnfk

pk=

K∑k=1

λ∗k ln gk

(recall that gk = fk/pk). Thus, in order to prove the desired inequality (50) it issufficient to verify the relation

K∑k=1

pkgk ln gk − ρK∑

k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri ≥ 0. (52)

If inequality (52) is strict, then (50) is strict as well.We have

K∑k=1

pkgk ln gk =K∑

k=1

fk lnfk

pk≥ 0, (53)

by virtue of the well-known inequality (recall that f, p ∈ ∆K)

K∑k=1

fk ln fk ≥K∑

k=1

fk ln pk, (54)

which is strict if(f1, ..., fK) 6= (p1, ..., pK). (55)

Therefore relation (52) is valid if

K∑k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri ≤ 0. (56)

In the rest of the proof, we will assume that the opposite inequality holds:

K∑k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri > 0. (57)

Then (52) will be obtained if we establish that

K∑k=1

pkgk ln gk ≥K∑

k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri. (58)

29

Indeed, then we have

K∑k=1

pkgk ln gk ≥K∑

k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri ≥ ρ

K∑k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri,

which yields (52). In the above chain of relations, the last equality holds by virtueof (57).

To verify inequality (58) we write

K∑k=1

pkgk ln gk =K∑

k=1

I∑i=1

λikr

igk ln gk =I∑

i=1

riK∑

k=1

λikgk ln gk,

andK∑

k=1

(ln gk)K∑

m=1

gm

I∑i=1

λimλi

kri =

I∑i=1

riK∑

k=1

K∑m=1

(λik ln gk)(gmλi

m).

Thus to prove (58) it remains to check that

K∑k=1

λikgk ln gk ≥ (

K∑k=1

λik ln gk)(

K∑k=1

gkλik) (59)

for each i = 1, ..., I.Let us fix i and put λk = λi

k. Inequality (59) can be written

E[g ln g] ≥ (E ln g)Eg,

where “E” stands for the weighted average

Eg =K∑

k=1

gkλk [λk > 0,K∑

k=1

λk = 1].

Observe that the function φ(g) = g ln g is strictly convex. Consequently,

Eφ(g) ≥ φ(Eg), (60)

and the inequality is strict if gk 6= gm for some k and m. Thus

E[g ln g] ≥ (Eg) lnEg ≥ (Eg)(E ln g), (61)

where the former inequality in this chain of relations coincides with (60) andthe latter is a consequence of the concavity of the function ln(·). Furthermore,both inequalities in (61) are strict provided that gk 6= gm for some k. If thelast condition does not hold, then fk/pk = c for some constant c, which mustnecessarily be equal to one because

∑fk =

∑pk = 1. Thus if gk = gm for all

k, m, then fk = pk, k = 1, 2, ...,K, which implies (see below) that pk = λ∗k for allk.

30

6th step. At the previous step of the proof, we established inequality (52) andhence (50). Moreover, the arguments conducted show that inequality (52) (andhence (50)) is strict if condition (55) is fulfilled. Indeed, if relation (56) holds then,under assumption (55), we have a strict inequality in (53), which implies a strictinequality in (52). Alternatively, if relation (57), opposite to (56), holds, thenstrict inequalities in (61) and (59) imply strict inequalities in (58) and (52).

Thus to complete the proof it suffices to show that if fk = pk, k = 1, 2, ...,K,then pk = λ∗k, k = 1, 2, ...,K. Indeed, if fk = pk, then we have

pk = ρK∑

m=1

pm

∑Ii=1 λi

mλikr

i

pm+ (1− ρ)λ∗k, k = 1, ...,K,

which implies

pk = ρK∑

m=1

I∑i=1

λimλi

kri + (1− ρ)λ∗k =

ρI∑

i=1

λikr

i + (1− ρ)λ∗k = ρpk + (1− ρ)λ∗k, k = 1, ...,K.

Thus (1− ρ)pk = (1− ρ)λ∗k, and so pk = λ∗k. �Proof of Proposition 4. The variables qk(s), pk and rk (k = 1, ...,K) are related

to each other by the system of equations (34). Suppose equations (30) hold. Then,from (34), we obtain:

λ∗m =K∑

k=1

[ρλ∗k + (1− ρ)Rk(s)]∑I

i=1 λimλi

kri

λ∗k, m = 1, ...,K,

or equivalently,

λ∗m =K∑

k=1

Rk(s)I∑

i=1

λimλi

kri

λ∗k, m = 1, ...,K, (62)

whereλ∗k = ERk(s), Rk(s) = ρλ∗k + (1− ρ)Rk(s).

Observe that if there are no redundant assets, the relation∑K

k=1 γkRk(s) = 0implies γ1 = ... = γK = 0. Indeed, suppose that

K∑k=1

γk[ρλ∗k + (1− ρ)Rk(s)] = 0. (63)

Then we have

0 = E

K∑k=1

γk[ρλ∗k + (1− ρ)Rk(s)] =K∑

k=1

γk[ρλ∗k + (1− ρ)λ∗k] =K∑

k=1

γkλ∗k,

31

which in view of (63) yields

K∑k=1

γkRk(s) = − ρ

1− ρ

K∑k=1

γkλ∗k = 0,

and so γ1 = ... = γK = 0 because the functions Rk(·), k = 1, ...,K, are linearlyindependent.

From formula (62) and the relation λ∗m = pm =∑I

i=1 λimri, we obtain

I∑i=1

λimri =

K∑k=1

Rk(s)I∑

i=1

λimλi

kri

λ∗k, m = 1, ...,K. (64)

We have∑K

k=1 Rk(s) = 1, and so equations (64) imply

K∑k=1

Rk(s)γmk = 0, m = 1, ...,K,

where

γmk =

1λ∗k

I∑i=1

λimλi

kri −

I∑i=1

λimri.

Since there are no redundant assets, we have γmk = 0 for each m and k. This gives

I∑i=1

λimλi

kri − λ∗k

I∑i=1

λimri = 0, k, m = 1, ...,K,

which can be written as

I∑i=1

λim(λi

k − λ∗k)ri = 0, k, m = 1, ...,K.

We derive two expressions from this equation. The first by setting k = m in theforegoing formula. The second by adding up over m = 1, ...,K. We find

I∑i=1

λik(λ

ik − λ∗k)r

i = 0, k = 1, ...,K, andI∑

i=1

(λik − λ∗k)r

i = 0.

Multiplying the second equation by −λ∗k and adding it up with the first, we obtain

0 =I∑

i=1

[λik(λ

ik − λ∗k)r

i − λ∗k(λik − λ∗k)r

i] =I∑

i=1

(λik − λ∗k)

2ri.

Consequently, we have

(λik − λ∗k)

2ri = 0, i = 1, ..., I, k = 1, ...,K. (65)

32

Suppose condition (a) holds. Since∑I

i=1 ri = 1 and ri ≥ 0, we have rj > 0 forsome j = 1, ..., I. Then, from (65), we get

λjk − λ∗k = 0, k = 1, ...,K, (66)

which is a contradiction.If condition (b) is fulfilled, then

∑Ii=2 ri = 1 − r1 > 0, and so rj > 0 for

some j = 2, ..., I. This implies (66), and the contradiction obtained completes theproof. �

Proof of Theorem 5. By virtue of Theorem 4, the limit r1∞ := lim r1

t exists a.s.and is strictly positive. Suppose the assertion we wish to prove is not valid. Thenwe have

P{0 < lim r1t < 1} > 0. (67)

Let us write for shortness Et(·) in place of E(·|st). If ξt is a non-positivesubmartingale, then Et−1ξt+1−ξt−1 → 0 a.s. (see the Lemma below). By applyingthis fact to the non-positive submartingale ξt = ln r1

t , we obtain

Et−1(lnr1t

r1t−1

+ lnr1t+1

r1t

) = Et−1 lnr1t+1

r1t−1

= Et−1ξt+1 − ξt−1 → 0 (a.s.). (68)

By using the fact that the random elements st−1, st and st+1 are independent andrepresenting the histories st, st+1 as

st = (st−1, st), st+1 = (st−1, st, st+1),

we get

Et−1(lnr1t

r1t−1

+ lnr1t+1

r1t

) = E[(lnr1t

r1t−1

)|st−1] + E[ln(r1t+1

r1t

)|st−1] =

∑s∈S

P{st = s} lnr1t (s

t−1, s)r1t−1(st−1)

+∑s∈S

P{st = s}∑σ∈S

P{st+1 = σ} lnr1t+1(s

t−1, s, σ)r1t (st−1, s)

=

∑s∈S

π(s) lnr1t (s

t−1, s)r1t−1(st−1)

+∑s∈S

π(s)∑σ∈S

π(σ) lnr1t+1(s

t−1, s, σ)r1t (st−1, s)

=

∑s∈S

π(s) ln f(s, rt−1(st−1)) +∑s∈S

π(s)∑σ∈S

π(σ) ln f(σ, rt(st−1, s)) =

∑s∈S

π(s) ln f(s, rt−1(st−1)) +∑s∈S

π(s)∑σ∈S

π(σ) ln f(σ, F (s, rt−1(st−1)). (69)

The last but one equality in the above chain of relations is valid because thestrategy of investor 1 coincides with the Kelly rule (λ1 = λ∗) and the last equalityholds because rt(st−1, s) = F (s, rt−1(st−1)).

33

By virtue of (67), (68) and (69), there exists a realization (s1, ..., st, ...) ofthe process of states of the world such that, for the sequence of vectors rt−1 =rt−1(st−1) ∈ ∆I , we have

0 < lim r1t−1 < 1, (70)∑

s∈S

π(s) ln f(s, rt−1) +∑s∈S

π(s)∑σ∈S

π(σ) ln f(σ, F (s, rt−1)) → 0. (71)

In the rest of the proof, we will fix such a realization (s1, ..., st, ...) and write rt−1

in place of rt−1(st−1).Since the simplex ∆I is compact, there exists a sequence t1 < t2 < ... and a

vector r ∈ ∆I such thatrtn−1 → r ∈ ∆I . (72)

It follows from (70) and (72) that the first coordinate r1 if the vector r = (r1, ..., rI)satisfies

0 < r1 < 1. (73)

Relations (71) and (72) imply∑s∈S

π(s) ln f(s, r) +∑s∈S

π(s)∑σ∈S

π(σ) ln f(σ, F (s, r)) = 0 (74)

because the function ln f(s, r) = ln g1(s, r) is continuous in r ∈ ∆I (see Proposi-tion 3).

By virtue of Theorem 3,∑s∈S

π(s) ln f(s, r) ≥ 0,∑σ∈S

π(σ) ln f(σ, F (s, r)) ≥ 0 (for all s ∈ S).

Consequently, it follows from (74) that∑s∈S

π(s) ln f(s, r) = 0, (75)

and ∑σ∈S

π(σ) ln f(σ, F (s, r)) = 0 for all s ∈ S. (76)

According to Theorem 2, relation (75) can hold only if

〈λk, r〉 = λ∗k, k = 1, ...,K, (77)

and equations (76) imply

〈λk, F (s, r)〉 = λ∗k, k = 1, ...,K, s ∈ S. (78)

By virtue of Proposition 4, relations (73), (77) and (78) cannot hold simultaneously.This is a contradiction. �

34

In the proof of Theorem 5 we used the following fact.

Lemma. Let ξt be a non-positive submartingale. Then the sequence of non-negative random variables ζt = Et−1ξt+1 − ξt−1 converges to zero a.s.

Proof. We have ζt ≥ 0 by the definition of a submartingale. Further, Eζt =(Eξt+1 − Eξt)+ (Eξt − Eξt−1), and so

N∑t=1

Eζt =N∑

t=1

(Eξt+1 − Eξt) +N∑

t=1

(Eξt − Eξt−1)

= EξN+1 − Eξ1 + EξN − Eξ0 ≤ −Eξ1 − Eξ0

because Eξt ≤ 0 for each t. Therefore the series of the expectations∑∞

t=1 Eζt ofthe non-negative random variables ζt converges, which implies (see, e.g., Corollaryto Theorem 11, in Chapter VI in [30]) that ζt → 0 (a.s.). �

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38

Review of Finance (2007) 11: 25–50doi: 10.1093/rof/rfm003

On the Evolution of Investment Strategiesand the Kelly Rule—A Darwinian Approach∗

TERJE LENSBERG1 and KLAUS REINER SCHENK-HOPPE2

1Norwegian School of Economics and Business Administration; 2University of Leeds

Abstract. This paper complements theoretical studies on the Kelly rule in evolutionary finance bystudying a Darwinian model of selection and reproduction in which the diversity of investmentstrategies is maintained through genetic programming. We find that investment strategies whichoptimize long-term performance can emerge in markets populated by unsophisticated investors.Regardless whether the market is complete or incomplete and whether states are i.i.d. orMarkov, the Kelly rule is obtained as the asymptotic outcome. With price-dependent ratherthan just state-dependent investment strategies, the market portfolio plays an important role asa protection against severe losses in volatile markets.

JEL Classification: G11, G12, C63

1. Introduction

In this paper, we pursue a Darwinian approach to the study of the evolutionof investment strategies in financial markets with short-lived assets. Themodel comprises the two main processes, selection and reproduction, in agenetic programming framework. According to this approach, center stageis occupied by the population which embodies the investment skills of manyindividual strategies. Our investors are simple-minded and unsophisticatedin the sense that they follow preprogrammed behavior rules which are theresult of mutations and crossovers. This simplicity is a key factor, as it opensup the possibility of studying, in quite a realistic context, the validity ofequilibrium predictions derived from theoretical models that impose strongassumptions on the market dynamics, as well as on individuals’ rationality orlearning behavior. Our approach complements these models by replacingtheir rationality assumptions with a Darwinian selection mechanism in

* We are most grateful for helpful and constructive comments from an anonymous refereeand from the editor, Peter Bossaerts. Thanks are also due to Jarle Møen for a helpfuldiscussion. Financial support from Leeds University Business School and the NationalCentre of Competence in Research ‘‘Financial Valuation and Risk Management’’ is gratefullyacknowledged.

The Author 2007. Published by Oxford University Press on behalf of the European Finance Association.All rights reserved. For Permissions, please email: [email protected]

26 TERJE LENSBERG AND KLAUS REINER SCHENK-HOPPE

which investment strategies emerge with a degree of risk aversion that isappropriate for survival. This approach also provides information on thestability properties of equilibria.

In the absence of a reproductive process that creates diversity, the marketselection dynamics for short-lived assets are well-studied from an evolutionaryperspective (Amir et al., 2005; Evstigneev et al., 2002 and 2006), as well as froma Bayesian viewpoint (Blume and Easley, 1992). The selection pressure in themodel considered here is provided by the wealth dynamics which give invest-ment strategies that accumulate more wealth than others a stronger impact onmarket prices and allocations. Kelly (1956) proved that if markets are com-plete and consist only of Arrow securities, the rule of ‘‘betting one’s beliefs’’eventually accumulates all wealth. This rule prescribes dividing wealth acrossassets according to the probability of their paying off. In incomplete markets,the Kelly rule generalizes to the strategy of setting portfolio weights equalto the assets’ expected relative payoffs (Hens and Schenk-Hoppe, 2005). TheKelly rule always has a non-negative growth rate relative to the market, and itsgrowth rate is strictly positive if the other investors do not hold the market port-folio, which yields zero relative growth. Moreover, if relative asset prices aregiven by the portfolio weights of the Kelly rule, this strategy corresponds to thelog-optimal investment (see Algoet and Cover, 1988; Hens and Schenk-Hoppe,2005). In summary, these papers find that the Kelly rule is selected by the mar-ket, in the sense of accumulating total market wealth in the long run. However,these results hinge on restrictive assumptions: either there is an investor whofollows the Kelly rule from the beginning, or there is a Bayesian learner withlogarithmic preferences whose prior includes the true model of the economy.

Related studies on market selection within general equilibrium modelsprovide less clear-cut results. In dynamically complete markets, the Bayesianlearner with the most accurate beliefs prevails in the long run, regardless of riskpreferences (Blume and Easley, 2006; Sandroni, 2000). Since only beliefs matterfor survival in complete markets, the Kelly rule does not play any prominentrole in this model. For incomplete markets, even the link between the accuracyof beliefs and survival does not hold in general. Blume and Easley (2006)provide an example in which an agent with wrong beliefs drives out a traderwith correct beliefs, even though the latter maximizes the logarithmic growthrate of wealth. The agent with incorrect beliefs judges returns too optimisticallyin relation to the true probability measure and ‘‘outsaves’’ the agent withcorrect beliefs. In complete markets, Pareto optimality of the equilibriumallocation precludes this from happening.

Research on the performance of rational versus irrational traders, as wellas the price impact of the latter, is also closely related to this paper becausetraders are characterized by rules of behavior. De Long et al. (1990 and 1992)show that noise traders can survive in markets and impact the prices in thelong run. Survival of a trader, however, is not a necessary condition for price

ON THE EVOLUTION OF INVESTMENT STRATEGIES AND THE KELLY RULE 27

impact, as clearly demonstrated by Kogan et al. (2006). They find that, even ifa trader’s wealth tends to zero, he can influence prices in the long run throughhis impact on the state-price density in states with low payoffs.

None of these results can be regarded as satisfactory from an applied pointof view. First, the general equilibrium approach to market selection does notoffer robust results on asset prices and their long-run dynamics. Since a traderwith an arbitrary (standard) utility function can dominate the market in thelong term, prices are not pinned down. Second, neither the general equilibriumapproach nor the existing literature on the Kelly rule allows for the entry ofnew investment strategies or the exit of unsuccessful ones. Third, one cannotconstruct the Kelly rule without sufficient information about asset payoffs andunderlying probabilities. Fourth, without this information it is not possible toconstruct an appropriate prior and Bayesian learning might fail. Finally, thechances of seeing the Kelly rule emerge in a market in which the traders donot have any knowledge of the theoretical results might be slim.

The approach used here provides an evolutionary finance model in whichthese shortcomings are overcome. We allow the set of investment strategiesto develop over time in a Darwinian fashion, using a genetic programmingalgorithm (Koza, 1992; Smith, 1980) to model the evolutionary process.Investment strategies are represented as computer programs, and newinvestment strategies are produced by genetic recombination of strategiesthat have performed well in the past. This maintains diversity in the pool ofinvestment strategies and occasionally produces new strategies with superiorcapability to generate wealth for those investors who adopt them.

Genetic programming belongs to a tradition in computer science whichemploys the principles of Darwinian evolution to breed artificially intelligentagents who can solve complex tasks (Holland, 1975). A pioneering applicationin finance is Arifovic (1996)’s analysis of exchange rate fluctuations in anoverlapping generations economy. Following Neely et al. (1997) and Allenand Karjalainen (1999), there is a growing literature on the usage of geneticalgorithms in identifying profitable trading rules in financial markets. Ina related paper, Lensberg (1999) analyzes a special case of Blume andEasley (1992)’s investment model in a genetic programming framework andfinds that surviving behavior rules indeed act as expected log-utility maximizerswith Bayesian updating.

Our model describes a situation where investors progressively improve theirskills without Bayesian rationality or other sophisticated learning procedures.This is possible because investor skill is tacit knowledge, acquired throughimitation and repeated trial-and-error, as emphasized by Polanyi (1967).Successful investors in our framework are like Friedman (1953)’s billiardplayers who somehow manage to get it right, but are unable to explain how.

Market clearing is modeled as in the market game of Shapley andShubik (1977). This allows us to focus on the strategic aspects of portfolio


choice while abstracting from details of implementation. Each agent has aninvestment strategy which is used to select portfolio weights. These weightsmay depend on information about the current state of the economy andon historical price information, but the agents must make their decisionwithout definite knowledge about the prices that prevail when their decisionis implemented in the market. Short selling is not permitted. This excludesone influence that is capable of correcting prices, although, as shown by DeLong et al. (1990 and 1991), this does not necessarily occur. In each newperiod, each agent carries over a portfolio of asset holdings from the previousperiod and receives state contingent dividends. The dividends are reinvestedin fresh assets, each of which becomes available in a fixed unit supply. TheShapley-Shubik mechanism clears the market by simply equating the marketcapitalization of each asset (its price) to the total wealth invested in it.

All of this is—by and large—faithful to the original setting proposed byKelly (1956), and yet it is in contrast to the general equilibrium approach.However, there is a close relation to the latter. Market game equilibriaconverge to competitive equilibria as the number of traders increases (Shapleyand Shubik, 1977), and any equilibrium price sequence can be mimickedwithin our framework when the portfolio weights are allowed to be time- andhistory-dependent (Amir et al., 2005).

We perform four sets of experiments in which genetic programs are suppliedonly with information about the current state of the economy. The experimentsdiffer with respect to the market structure and the fundamental stochasticprocess. The findings are positive throughout. The Kelly rule emerges from thepopulation of genetic programs in all experiments. Throughout a transitoryperiod, however, the market is closer to the theoretical equilibrium thanare the individual investment strategies, an observation that is in line withexperimental findings (see Bossaerts et al., 2005). The information spreadsefficiently through the population so that, eventually, the majority of thepopulation follows the Kelly rule, i.e., convergence of market and strategiesprevails. In this sense the optimal investment strategy is indeed learned. Thefindings are robust with respect to the level of noise that is generated bymutation, crossovers and the inflow of wealth.

The impact of the availability of price information is studied in detailwithin one of these experiments. In that setting, the Kelly rule coincides withthe growth-optimal investment, which is independent of market prices eventhough it depends on the state. From a genetic programming perspective, theproblem becomes considerably harder to solve because the number of inputsto the decision problem increases from 2 (state and asset) to 5 (state, asset andthree prices). Market prices and strategies still converge to those predicted bythe Kelly rule, but the transitional dynamics and successful investment stylesare qualitatively different. An analysis of wealthy genetic programs showsthat the market portfolio plays an important role. Purchasing the market


portfolio ensures that a constant share of the total wealth is maintained andlow levels of wealth are avoided. This increases a strategy’s chance of survivalby reducing the probability of being deleted by the genetic recombinationprocess. If state-contingent market clearing prices are perfectly anticipated,such a strategy neither loses nor gains in the market dynamics. In volatilemarkets, there is slippage, but even an approximate market portfolio providesconsiderable downside protection.

This study is numerical, and one might ask whether analytical resultsexist that would put more solid ground under our findings. From amathematical perspective, the major challenge—and also the main departurefrom the model considered in Amir et al. (2005)—arises from endowingnewly created strategies with wealth. This process depends on the stateof the market rather than being an exogenous and purely stochasticevent. Given that the analysis in Amir et al. (2005) is quite sophisticated,analytical results do not appear to be straightforward. Moreover, Amiret al. (2005) and related papers by Blume and Easley (1992) and Evstig-neev at al. (2002) only provide results on the asymptotic dynamics of the basicmodel. They neither study transient behavior nor provide estimates on thespeed of convergence for the short- and medium-term. Our paper, however, isparticularly concerned with these two issues. Particular emphasis is placed onthe properties of the wealthiest investment strategy where little or nothing isknown, in general, because of its highly path-dependent nature.

The remainder of the paper is organized as follows. Section 2 presentsthe Darwinian model of a financial market in three steps: wealth dynamics,2.1, investment strategy dynamics, 2.2, and implementation of the geneticalgorithm, 2.3. The experiments are presented in Section 3: General resultsare given in 3.1, and a detailed analysis of the complete market with Markovstates is provided in 3.2. The case of price-dependent strategies is consideredin Section 4. Section 5 concludes.

2. A Darwinian Finance Model

This section introduces an evolutionary model of a financial market with short-lived assets. The model incorporates the two Darwinian processes of selectionand reproduction. Selection is given by the wealth dynamics among a fixed setof investment strategies, and reproduction imposes a dynamic structure on theset of strategies itself. Both processes are captured here by implementing themodel using genetic programming.

2.1 WEALTH DYNAMICS

We briefly recall the evolutionary finance model studied in Amir et al. (2005),Evstigneev et al. (2002 and 2006) and Hens and Schenk-Hoppe (2005). This


model describes the wealth dynamics among a given pool of I investmentstrategies that interact in a financial market. Time is discrete, t = 0, 1, 2, . . ..There are K assets with random payoffs Dk(s) ≥ 0, k = 1, . . . ,K, with∑K

k=1 Dk(s) > 0. Here s = 1, . . . , S denotes the state of nature. Each assetis short-lived and in fixed supply of one unit. An investment strategy isa sequence of time- and history-dependent vectors of portfolio weightsλi

t = (λi1,t , . . . , λi

K,t ), λik,t ≥ 0 and

∑Kk=1 λi

k,t = 1.The price of asset k at time t is given by qk,t := λk,twt = ∑I

i=1 λik,t wi

t , i.e., qk,t

is equal to the total amount of wealth invested in asset k. Investor i’s portfolioholdings in asset k is determined as λi

k,t wit /λk,twt = λi

k,t wit /qk,t which is a

fraction of the one unit supplied. Each strategy’s wealth in the next period isdetermined by the total receipts of random asset payoffs which are distributedaccording to the portfolio holdings.

The evolution of the distribution of wealth wt = (w1t , . . . , wI

t ) acrossinvestment strategies is governed by

wit+1 =

K∑k=1

Dk(st+1)λi

k,t wit

λk,twt

(1)

for i = 1, . . . , I . The state st+1 is randomly drawn according to a givenprobability distribution.

The pricing equation qk,t = λk,twt merits a more detailed discussion. Itis the market clearing condition of the Shapley and Shubik (1977) marketgame, which equates the market capitalization of an asset to the total amountof wealth invested in it. Since each asset is in one-unit fixed supply, thisis simply the price of asset k. Thus, the Shapley-Shubik market gamesimultaneously clears, with each time step, K markets and yields a uniqueshort-term equilibrium price vector. That is quite different from agent-basedmodels, where usually only one market-clearing price is needed (Hommes,2001). From the definition, it is clear that each strategy’s impact on the priceis proportional to its wealth. Short selling is excluded to avoid bankruptcy,which would be prevalent in the presence of a short-run equilibrium. ForArrow securities, which have a positive payoff in one state of the worldand pay zero otherwise, the price determines the odds of the correspondingbet; they are given by Dk(st+1 = k)/qk,t . In this respect, the market-clearingmechanism corresponds to the one used in parimutuel betting markets.

Equation (1) can be interpreted as the market selection dynamics. Strategiesthat have higher wealth than their competitors are considered to be fitter. Ifone strategy accumulates total wealth in the long term, it is said to be selectedby the market. Since prices are a weighted combination of the strategies withweights equal to wealth, such a strategy asymptotically ‘determines’ assetprices: relative prices are asymptotically equal to the portfolio weights of aselected strategy.


The introduction of investment strategies, as well as the above definitionof asset prices, marks a departure from the general equilibrium approach tomarket selection, where agents have demand functions and maximize utilityover an infinite time-horizon (Blume and Easley, 1992 and 2006; Sandroni,2000). Notwithstanding the apparent simplicity of the model, Amir et al. (2005)have shown that each equilibrium price path in such a general equilibriummodel can be obtained by an appropriate specification of investment strategieswhose portfolio weights are, in general, time- and history-dependent.

2.2 DYNAMICS OF INVESTMENT STRATEGIES

The wealth dynamics of the standard evolutionary finance model describehow the aggregate behavior changes over time for a fixed set of investmentstrategies. Here, the set of investment strategies is changing over time as well.This allows investigating whether the market mechanism is strong enough forthe population to discover the Kelly rule, even though each individual investorlacks the analytical ability to do so.

The maximum number of investment strategies that can be active in themarket at any period of time is limited to a finite number I . Each investmentstrategy λi = (λi

1, . . . , λiK) is represented by a program which is given in the

form of a function

λi

: S × {1, . . . ,K} → R, (2)

where S is the set of potential signals σ associated with the true state s. λi

k(σ )

is the non-normalized budget allocated to the purchase of asset k, given thatthe last observed signal is σ . Examples of programs are provided in Table I . Inorder to compute the portfolio weights for a given function λ

i, a normalization

is carried out as follows. First define λi

k(σ ) := max{0, λi

k(σ )}, and then letλi

k(σ ) := λi

k(σ )/[∑K

n=1 λi

n(σ )]. If the denominator is zero, we set λik(σ ) = 1 for

some randomly chosen k.Two main cases are considered: (a) state-dependent strategies with complete

information about states, i.e., S = {1, . . . , S}, and (b) price-dependentstrategies where the set of signals contains the last observed price vectorcorresponding to the present state, i.e., S = {1, . . . , S} × R

K . In general theinformation set could be an indexed partition of the state space or containother information such as asset payoffs and their moving averages.

2.3 IMPLEMENTATION BY GENETIC PROGRAMMING

Genetic programming (GP) (Koza, 1992; Smith, 1980) is a technique forprogramming computers by natural selection. It addresses the challenge ofgetting a computer to do what needs to be done without explaining everything


in detail, and it attempts to achieve this goal by breeding a population ofprograms using the principles of Darwinian selection and reproduction. Theidea of mimicking evolution to simplify software design was introduced byHolland (1975), and it has been used successfully by computer scientists tosolve problems in a variety of engineering fields. For our purpose, it is used asa model of decision making in the presence of tacit knowledge and boundedrationality.

GP-algorithms produce programs which consist of instructions to read ormanipulate data. Each program produces some output, which the modelerinterprets as the action taken by the program, and this action is evaluatedto obtain a measure of the program’s fitness. In our particular context, theaction taken consists of the portfolio weights allocated to the K assets and thenatural fitness measure is the accumulated wealth of the program.

In order to find the computer program which best solves a given task, theGP-algorithm starts by randomly generating a large population of programs.It then continues for a large number of iterations by replacing low performingprograms with a genetic recombination of high performing ones. The standardgenetic operators are crossover and mutation. These are explained below, alongwith an additional operator (noise) that is used here to test the robustness ofour results.

GP-algorithms differ in the way they mimic Darwinian evolution. Here, asteady-state algorithm with tournament selection is used. It works as follows:

1. Tournament: Randomly select four programs from the pool and rankthem according to their accumulated wealth.

2. Reproduction: Replace the two programs with lowest wealth with copiesof the other two.

3. Mutation: Each of the two programs copied undergoes a mutation withprobability µ: randomly select a single instruction from the program,and replace it with a randomly generated instruction.

4. Crossover: With probability χ , recombine the genetic material of thetwo copied (and possibly mutated) programs by swapping one randomlyselected set of instructions from both programs.

5. Noise: Each of the two newly generated programs is replaced by arandomly selected program with probability η.

If a program has strictly positive wealth, it leaves the tournament with thesame wealth. However, if a program has zero wealth, it is endowed with onepercent of the average wealth which is given by (1/I)

∑Kk=1 Dk(st+1).

The algorithm is run for a population size of 1,000 and 250,000 iterations,with 20 tournament selections per iteration. The mutation and crossoverprobabilities are set to µ = 0.9 and χ = 0.5, respectively, and the noiseprobability η is varied from 0 to 0.96 in order to check the robustness of ourresults.


GP-algorithms also differ with respect to the type of building blocks they useto construct individual programs. One typically uses a subset of elementaryinstructions from some existing programming language, such as LISP, Javaor machine code. In this paper, we use a machine code version of GP, whichis introduced by Nordin (1997): each program consists of a list of machineinstructions which operate on variables and constants stored in memory, usingthe CPU floating point registers to store and manipulate temporary variables.

Table I illustrates some aspects of the machine code GP algorithm bymeans of an example in which the maximum program length is limited to 6instructions. In practice, the maximum program length is much larger; in oursimulations, it consists of 128 instructions. The left part of the table depicts twoprograms, A and B, with 5 and 6 instructions, respectively. The right part showsthe outcome of a crossover at instruction slots 3–6, which produces two newprograms, C and D. R0 and R1 refer to floating point registers 0 and 1 of an Intelcompatible CPU, which has a total of 8 such registers. The GP algorithm clearsthese registers by loading them with the value 0.0 before passing a program tothe CPU for execution. Input variables consist of the state s and asset k (seeEquation (2)), and the output is the content of register R0 after all programinstructions have been processed by the CPU. The output is interpreted as thenon-normalized portfolio weight λk(s) for asset k in state s. For Program A,the normalized portfolio weights λk(s) are constant and equal to 1/K for eachstate and asset, while for Program C they vary according to state and asset.

Instead of using GP to model the dynamics of investment strategies, a geneticalgorithm (GA) (Holland, 1975) could be used in the first set of experiments.The main difference is that GA operates on vectors of numbers instead ofvectors of program instructions. GA is applicable as long as the set of signalsis finite (e.g., if only state and asset index are contained in the informationset). Any investment strategy can then be represented as a vector of realnumbers. Evolving such vectors with GA is simpler and computationally lessdemanding than evolving functions with GP. If strategies can also use past

Table I. Example of the program structure and the crossover operation

Instr.Before crossover After crossover

slot Program A Program B Program C Program D

1 R0 = s R1 = k R0 = s R1 = k

2 R1 = k R0 = R0 − 2 R1 = k R0 = R0 − 2

3 R1 = R1 ∗ R0 R0 = R0 ∗ R0 R0 = R0 ∗ R0 R1 = R1 ∗ R0

4 R0 = R0/5 R0 = R0/R1 R0 = R0/R1 R0 = R0/55 Return R0 R0 = min(R0, R1) R0 = min(R0, R1) Return R0

6 Return R0 Return R0

λk(s) s/5 min(4/k, k) min(s2/k, k) −2/5


prices, the information set is a continuum and so is the range of strategies. Forthese experiments, the full generality of GP, which provides us with functionalrelations, is needed to analyze the model.

3. The Experiments

Four sets of experiments are simulated to analyze the impact of a change inthe market structure, as well as in the stochastic process that determines thestate. The market is either complete (i.e., the rank of the payoff matrix is equalto the number of states) or incomplete (i.e., there are fewer assets than states).The state of nature is given either by an i.i.d. process or a Markov process.

In each experiment, there are three states of nature, s = 1, 2, 3, and K = 3resp. K = 2 assets. The payoff matrix D is defined in Table II. Note that thecomplete market consists of Arrow securities.

In the i.i.d. case, all states have equal probability, i.e., πs = 1/3 for s =1, 2, 3. In the Markov case, however, the probability of the next period’s statedepends on the current state. The matrix of transition probabilities is given by

� = 0.7 0.2 0.1

0.1 0.7 0.20.2 0.1 0.7

. (3)

The stationary distribution of this Markov process, denoted by ρ, is given byρ1 = ρ2 = ρ3 = 1/3.

The Kelly rule in each of these cases is given by the expected values of theassets’ relative payoffs:

λ∗k(s) =

S∑u=1

�suRk(u), where Rk(u) = Dk(u)∑K

n=1 Dn(u), (4)

with k = 1, . . . ,K and s = 1, . . . , S. In the i.i.d. case, the current statedoes not impact the probability of the next period’s state, i.e., �su ≡ πu,and, therefore, the Kelly rule is a constant vector. Table III summarizes thenumerical values of the Kelly rule in the four experiments.

The investment strategy λ∗ provides the equilibrium prediction for thelong-run outcome of individual behavior, as well as for the asymptotic values

Table II. Payoff matrix D in the complete, resp. incomplete, marketcase

Complete market Incomplete market

D = 1 0 0

0 2 00 0 3

D =

1 1

1 10 3


Table III. Kelly rule in the four experiments

IID Markov

Complete Incomplete Complete Incomplete

λ∗ ≡(

13,

13,

13

)λ∗ ≡

(13,

23

) λ∗(s = 1) = (.7, .2, .1) λ∗(s = 1) = (.45, .55)

λ∗(s = 2) = (.1, .7, .2) λ∗(s = 2) = (.40, .60)

λ∗(s = 3) = (.2, .1, .7) λ∗(s = 3) = (.15, .85)

of (relative) asset prices. The two main questions to be investigated are (1)whether the competitive process of genetic programming is powerful enoughto drive the population towards the Kelly rule and, if yes, (2) whether thisconvergence is robust against noise.

To answer the first question, the distance between the Kelly rule and thewealthiest investment strategy is measured, as well as the distance to the marketprices. Prices are equal to the average strategy of the population because theycorrespond to the wealth-weighted average, as explained in Section 2.1. Thewealthiest strategy is defined here as the sequence of investment strategies thatis provided by selecting, in any one period in time, the strategy with the highestwealth. Two measures are applied: (a) the Euclidean distance and (b) theexpected growth rate. The latter quantity measures the growth potential of aninvestment strategy relative to a benchmark. Two benchmarks are employed:the Kelly rule and the current state of the market, which is given by the currentprice system.

The second question on the robustness of the convergence is analyzed bycomparing long-term outcomes for different values of the average numberof randomly generated programs. This is achieved by varying the noiseprobability η, see Step 5 of the GP-algorithm.

3.1 SIMULATION RESULTS

Each experiment consists of a simulation of a population of I = 1, 000 geneticprograms. Programs are initialized by randomly chosen functions, which arearrays of random length, filled with random draws from the instruction set.The noise probability η is varied, in each experiment, between 0 and 0.96 withincrements of 0.04. The Darwinian finance model is run for a total of 250,000periods for each set of parameters.

The distance measures are defined as follows. The probability-weightedEuclidean distance between a strategy λi and the Kelly rule λ∗ is given by

d∗(λi) := ∑Ss=1 ρs

√∑Kk=1(λ

ik(s) − λ∗

k(s))2. (5)

In the i.i.d. case, the stationary distribution ρ is given by ρs = πs .


The second distance measure can be defined in terms of the growth rate of astrategy λi relative to the Kelly benchmark, g∗(λi), or its growth rate relativeto the market, gM(λi). The first one is defined as

g∗(λi) := exp

(S∑

s=1

ρs

S∑u=1

�su ln

(K∑

k=1

Rk(u)

λ∗k(s)

λik(s)

)), (6)

where the relative payoffs Rk(u) are given by (4) and ρ is the stationarydistribution of the stochastic process that determines the state of nature. Inthe i.i.d. case, one has �su ≡ πu and λ∗

k(s) ≡ λ∗k in Equation (6). Taking the

exponential allows a comparison of growth rates in terms of percentages.The standard definition is obtained by dropping the exponential function (seeHens and Schenk-Hoppe, 2005). One could also try to measure the distanceby the relative entropy of strategy λi and the Kelly rule. In the case of Arrowsecurities the relative entropy is equal to the expected logarithmic growthrate of λi . However, if the market is incomplete, the relative entropy, ingeneral, provides no information about a strategy’s growth rate (see, e.g.,Blume and Easley, 2006; Sandroni, 2005). The growth rate relative to themarket, gM(λi), is defined by replacing λ∗

k(s) in (6) by the (moving) benchmarkλM

k (s) := pk(s) = qk(s)/[∑K

n=1 qn(s)]. pk(s) is the relative price of asset k.The simulation results for these different measures are summarized in

Tables IV and V. In both IID experiments, investment behavior convergesquickly to the Kelly benchmark for all noise levels. This can be seen from

Table IV. Summary statistics for 4 sets of experiments without price information

The table contains sample statistics across all runs and iterations for relative growth ratesand distances from the Kelly benchmark. λW denotes the wealthiest strategy and λM

denotes the market portfolio. The number of observations for each variable is 6,250 (250samples of 25 runs).

IID Markov

Variable Mean Std.dev Min Max Mean Std.dev Min Max

Complete market

d∗(λW ) 0.001 0.022 0.000 0.817 0.054 0.048 0.003 1.128d∗(λM) 0.001 0.006 0.000 0.305 0.053 0.044 0.002 0.715g∗(λW ) 0.999 0.038 0.000 1.000 0.988 0.033 0.000 1.000gM(λW ) 0.998 0.044 0.000 1.005 0.996 0.058 0.000 1.008

Incomplete market

d∗(λW ) 0.003 0.009 0.000 0.196 0.061 0.032 0.014 0.374d∗(λM) 0.003 0.007 0.000 0.179 0.023 0.017 0.001 0.251g∗(λW ) 1.000 0.000 0.990 1.000 0.999 0.001 0.959 1.000gM(λW ) 0.999 0.031 0.000 1.009 0.989 0.098 0.000 1.010


Table V. Average wealth of 5 groups of investment strategies in the 4 sets of experimentswithout price information

This table shows the proportion of wealth owned by the wealthiest strategy (λW ) and bythose strategies that are close to the Kelly rule in terms of relative growth rates (g∗ ≥ . . .).Numbers are averages taken across 6,250 observations for each experiment.

Dividend MarketλW g∗ ≥ 0.950 g∗ ≥ 0.975 g∗ ≥ 0.990 g∗ ≥ 0.999

process structure

IIDComplete 0.040 0.999 0.999 0.998 0.998Incomplete 0.789 0.997 0.997 0.995 0.991

MarkovComplete 0.568 0.981 0.926 0.538 0.217Incomplete 0.528 0.996 0.994 0.988 0.625

the small means and standard deviations of the two distance measures d∗(λW)

and d∗(λM) in Table IV. In the Markov experiments, the mean distancesfrom the Kelly benchmark are higher. Nonetheless, the wealthiest strategyperforms quite well relative to the market, as well as to the Kelly benchmark,as manifested by the high relative growth rates g∗(λW) and gM(λW). Table Vshows, however, that the wealthiest strategy does not possess the entire marketwealth. A substantial amount of wealth is managed by other investmentstrategies (for instance nearly 50% in both Markov experiments), but thosestrategies are also quite close to the Kelly rule in terms of relative growth rates,as can be seen from the 4 rightmost columns of Table V.

In the Markov cases, particularly in the incomplete market setting, the pricesystem is closer to the Kelly benchmark than the wealthiest investment strategy,i.e., the market is ‘‘smarter’’ than the most successful investor, cf. Table IV.This finding is in line with Bossaerts et al. (2005). In experiments with humansubjects, they find that, while asset prices converge quickly and agree—byand large—with those of the CAPM, the portfolio choice predictions of thistheory remain significantly off target throughout the experiment.

We analyze the Markov experiments in more detail by regressing distanceson noise and simulation time in a panel data model with random effects andautoregressive errors. The estimated relationship is

ln(d∗rt ) = β0 + β1 ln(t/250) + β2 ln(1 − ηr) + ur + εrt , (7)

where ur is a run specific error term and εrt is AR(1). d∗rt is the Euclidean

distance from the Kelly rule at iteration 1000 · t of run r and ηr is the noiselevel in run r . Table VI contains the results. Estimated parameters are givenin Panel (A) and predicted distances for selected values of the explanatoryvariables are given in Panel (B).

The negative coefficients of ln(t/250) and ln(1 − ηr) in panel (A) showthat distances from the Kelly rule decrease over time and that the processslows down when the noise level is increased. Panel (B) gives the predicted


Table VI. Model and predicted values of Euclidean distance from the Kelly rule.Markov dividend process.

The estimated model is ln(d∗rt ) = β0 + β1 ln(t/250) + β2 ln(1 − ηr) + ur + εrt , where

ur is a run specific error term and εrt is an AR(1) process. d∗rt is the Euclidean distance

from the Kelly rule at iteration 1000 · t of run r and ηr is the run r noise level. λW

denotes the wealthiest strategy and λM denotes the market portfolio. Standard errorsare given in parentheses. * and ** denote significantly different from zero at the 5% and1% level, respectively (two-tailed test).

Dependent(A) Parameters (B) Predicted distance

variable β0 β1 β2 R2 t\η 0.9 0.5 0.0

Complete market

d∗(λW )−4.238** −0.585** −0.490** 0.474 5 0.439 0.200 0.142(0.139) (0.019) (0.113) 250 0.045 0.020 0.014

d∗(λM)−4.220** −0.541** −0.502** 0.477 5 0.387 0.172 0.122(0.145) (0.018) (0.119) 250 0.047 0.021 0.015

Incomplete market

d∗(λW )−3.246** −0.170** −0.190* 0.196 5 0.117 0.087 0.076(0.109) (0.006) (0.090) 250 0.060 0.044 0.039

d∗(λM)−4.515** −0.254** −0.323** 0.284 5 0.062 0.037 0.030(0.073) (0.008) (0.059) 250 0.023 0.014 0.011

(shrinking) distance to the Kelly rule for noise levels 0, 0.5 and 0.9. The moststriking observation is that, in all cases, the wealthiest investment strategy, aswell as the market, converges to the Kelly rule. Even for high values of thenoise parameter, convergence prevails, though at a lower speed.

Panel (B) of Table VI also shows that the market is closer to the Kelly rulethan the wealthiest investment strategy, in particular at the beginning of eachexperiment. The effect is particularly pronounced in the incomplete marketcase with Markovian states. This observation implies that there are strategiesin the market that move prices closer to the Kelly rule, though they do notprovide the wealthiest strategy in the market.

3.2 COMPLETE MARKET WITH MARKOV STATES

This section provides a more detailed analysis of the case with a completemarket and a Markov payoff process. The goal of this exercise is to obtain abetter insight into the mechanisms that drive the Darwinian dynamics whichgive rise to the findings reported above. This particular case is chosen becauseit exhibits the highest deviation of prices from the Kelly rule. The noiseprobability is set to 20% throughout the following.


Figure 1. Euclidean distance between the market prices resp. the wealthiest strategy andthe Kelly rule.

Figure 1 shows the distance between the Kelly benchmark and the market,d∗(λM), as well as the wealthiest strategy, d∗(λW), for one run of the simulation,see (5). Both distance measures decrease almost monotonically, and the descentmainly follows a step function. In this leapfrogging movement towards thebenchmark, the wealthiest strategy is overtaken, every now and again, by abetter performing competitor. This occurs only nine times during the timehorizon considered in Figure 1. This observation implies that, most of thetime, the wealthiest strategy is not the rule with the highest growth rate. Thecompeting strategy that eventually replaces the wealthiest rule is typicallymuch closer to the Kelly rule, as shown by the size of the steps in theconvergence. If the wealthiest strategy is close to the Kelly rule, this processtakes a considerable amount of time. The reason is that genetic programmingcreates better, but somewhat poorer, strategies from the genetic material ofthe wealthiest rule rather than improving the currently richest strategies. After250,000 iterations, the wealthiest strategy is very similar to the Kelly rule, witha distance of only 0.003. This number is in agreement with the results reportedin Table VI.

The distance between market prices and the Kelly rule displays similarbehavior though, in most periods, this measure is closer to the Kelly rulethan the wealthiest strategy. Only in the last 20,000 iterations is the marketprice further away from the benchmark than the wealthiest strategy. Thisparticular effect is caused by the relatively small perturbation through thenoise component of the tournament, which has a stronger impact when thepopulation is very close to the benchmark.

On the aggregate level, interest focuses on the convergence dynamics of assetprices. Figure 2 shows the relative price of Asset 1 for each period of time andfor each state s = 1, 2, 3 (one of which is revealed in any given period) for a


Figure 2. Relative price of Asset 1 by state and iteration. Horizontal lines correspondto the Kelly benchmark.

simulation of one run of 250,000 iterations. The benchmark, which is given bythe Kelly rule, is the set of relative asset prices p∗

1(1) = 0.7, p∗1(2) = 0.2 and

p∗1(3) = 0.1 in states 1, 2 and 3, respectively. The simulated prices are some

distance from the benchmark in the first 150,000 iterations, but they exhibita clear tendency to converge to this benchmark. After approximately 175,000iterations, the relative prices are almost identical to those derived from theKelly rule. However, systematic mispricing occurs over long time horizons.For instance, in state 1, the asset is first undervalued and later overvalued.Moreover, the prices in all states can simultaneously be too low comparedto the benchmark. In Figure 2, this occurs during iterations 30,000–40,000.Relative prices of an asset do not necessarily sum up to one across states, butof course, the sum across relative asset prices is equal to one in each state.

A proxy for the genetic material that is present in the population is providedby the distribution of growth rates relative to the Kelly rule. These valuesare, in fact, only potential growth rates because many strategies have zerowealth. Figure 3 reports the distribution of the individual growth rates relativeto the Kelly rule for the entire population across iterations. An analysis ofthe distribution of growth rates relative to the current market prices gives asimilar picture to the one provided in Figure 3.

Programs with a zero relative growth rate (defined in (6)) consist of strategiesthat are not fully diversified; a behavior that carries a positive probability oflosing all one’s wealth. The second group consists of strategies with positiverelative growth rates below 0.95. A large proportion of these strategies haveconstant and equal portfolio weights, a strategy which yields a relative growthrate of 0.7432. These programs simply return the same constant for eachasset and state. They are quite successful in the first 1,000 iterations ofthe simulation, because they avoid mistakes, such as under-diversification


Figure 3. Distribution of growth rates in the population relative to the Kelly rule. (Dataare exponentially smoothed, with a parameter value of 0.2.).

or extreme deviations from actual probabilities. Since the probability ofgenerating such a strategy from scratch or by crossover is high, the populationcontains a large proportion of those in the early stages of the simulation.

The distribution of relative growth rates changes significantly over time. Thenumber of strategies with equal weights is quickly reduced from an initiallyhigh level. The reason is that, as the fraction of efficient strategies increases,the equal weight strategies lose money faster, which reduces their chances ofreproducing. Halfway into the simulation, strategies with relative growth ratesabove 0.999 begin to take over and, after 250,000 iterations, they make upabout 40% of the population. This illustrates the ability of the GP-algorithmto generate extremely efficient investment strategies.

Interestingly, the fraction of strategies with a zero relative growth rateincreases from approximately 20% to 30% during the simulation. Since thesestrategies are not fully diversified, they act like gamblers who take extremepositions in some assets. We believe this result is driven by the followingevolutionary mechanism: since wealth is a prerequisite for reproduction, thestruggle for survival is basically a struggle to get rich. As the populationbecomes dominated by extremely efficient strategies, it grows harder to getrich by investing prudently. This makes gambling for large stakes increasinglyattractive as it can open a window of opportunity to reproduce. By thismechanism, competition produces not only more winners but also morelosers, which is in agreement with Figure 3.

4. Price-dependence and Market Dynamics

This section presents a generalization of the previous setting: strategies aregiven access to information on past prices. The main issue is how and to what


extent genetic programs use this additional information. To obtain robustresults, a case is analyzed in which the growth-optimal portfolio coincides withthe Kelly rule for all prices, but where the latter is dependent on the state ofnature. Prices should not matter for a growth-oriented investor, though statesshould. We focus on the experiment with a complete market and a Markovdividend process, studied in detail for state-dependent strategies in Section 3.2.

The availability of price information indeed gives rise to new types ofbehavior. The simulation results reported below show in particular that aprominent role is played by the market portfolio. Its importance is due to thefact that buying the market portfolio entails a constant share of total wealthand hence increases its probability of survival and reproduction relative to astrategy with more volatile payoffs.

In order to see that buying the market portfolio preserves relative wealth,consider a strategy, denoted λM , whose portfolio weights correspond to therelative prices pk,t (which are the market portfolio weights):

λMk,t = pk,t := qk,t∑K

n=1 qn,t

. (8)

Let wMt denote the wealth of strategy λM . The portfolio holdings of this

strategy are equal to

θMk,t := λM

k,twMt

qk,t

= wMt∑K

n=1 qn,t

= wMt

Wt

= rMt ,

where Wt := ∑Ii=1 wi

t = ∑Kn=1 qn,t is the total wealth and rM

t is the relativewealth of strategy λM . It holds the same amount of units of each asset becauseθM

k,t is independent of k. Its payoff can be computed as

wMt+1 =

K∑k=1

Dk(st+1)θMk,t =

K∑k=1

Dk(st+1)rMt = D(st+1)r

Mt ,

with D(st+1) denoting the aggregate dividend payoff in state st+1. It followsfrom (1) that D(st+1) = Wt+1. Thus, an investor who buys the market portfoliohas constant relative wealth, i.e., rM

t+1 = rMt .

To introduce price dependency, an investment strategy is defined here asa function λ : {1, . . . , S} × R

K × {1, . . . ,K} → R, where λk(s, p(s)) is the(non-normalized) portfolio weight of asset k given that the current state is s.The information set is S = {1, . . . , S} × R

K . The new argument p(s) denotesthe most recently observed relative price vector corresponding to the currentstate s. For instance, if (st−3, st−2, st−1, st ) = (1, 1, 2, 1), where s = st = 1, thenp(s) is the vector of relative prices observed in period t − 2. Trading is subjectto slippage because strategies only have access to the last observed price vector(i.e., for the current state) rather than the current market clearing prices. This


slippage comes from the time elapsing from order placement to execution,which precludes the purchase of a perfect market portfolio. However, if pricesexhibit low volatility, the strategy defined in (8) delivers an asset allocationthat corresponds well with the market portfolio. In periods of high volatility,the fit is not so good. Despite the slippage, this proxy of the market portfolioprovides an opportunity to protect one’s investment from downside risk at thecost of missing out on its upside. Carrying out a market portfolio investmentis actually a simple task for the genetic programs: they only have to output theprice of the relevant assets, whereas any other behavior requires computation.The following shows that successful investment strategies make skillful use ofthis investment opportunity.

Extending the information set from states to states and prices more thandoubles the number of input variables to any given strategy. The new data arevectors of real numbers rather than elements of a finite set. This increases thecomplexity faced by the GP-algorithm considerably. Since this is likely to slowdown convergence, the effect of doubling the population size to 2,000 is tested.Table VII presents the result of applying the regression model in Equation (7)to data from a simulation with 25 runs, with noise levels between 0 and 96%,and population sizes of 1,000 and 2,000. The increase in population size indeedcompensates for the increase in complexity, cf. Table VI. As in the correspond-ing base case considered in Section 3.2, the noise probability is set to 20%.

Table VII. Model and predicted values of Euclidean distance from the Kelly rule for price-dependent strategies. Markov dividend process and complete market.

The estimated model is ln(d∗rt ) = β0 + β1 ln(t/250) + β2 ln(1 − ηr) + ur + εrt , where ur is a run

specific error term and εrt is an AR(1) process. d∗rt is the Euclidean distance from the Kelly rule

at iteration 1000 · t of run r and ηr is the run r noise level. λW is the wealthiest strategy and λM

is the market portfolio. Standard errors are given in parentheses. * and ** denote significantlydifferent from zero at the 5% and 1% level, respectively (two-tailed test).

Dependent(A) Parameters (B) Predicted distance

variable β0 β1 β2 R2 t\η 0.9 0.5 0.0

Population size of 1,000

d∗(λW )−3.354** −0.340** −0.239** 0.259 5 0.229 0.156 0.132(0.074) (0.011) (0.060) 250 0.061 0.041 0.035

d∗(λM)−3.360** −0.336** −0.241** 0.253 5 0.226 0.153 0.129(0.074) (0.011) (0.060) 250 0.061 0.041 0.035

Population size of 2,000

d∗(λW )−3.911** −0.482** −0.334** 0.392 5 0.285 0.166 0.132(0.067) (0.013) (0.054) 250 0.043 0.025 0.020

d∗(λM)−3.934** −0.484** −0.338** 0.388 5 0.283 0.164 0.130(0.068) (0.013) (0.055) 250 0.043 0.025 0.020


The dynamics of the Euclidean distance between the wealthiest strategy andthe Kelly rule, reported in Figure 4, are strikingly different from the basecase depicted in Figure 1. Volatility is much higher, there is no leap-froggingeffect and, despite the variability of prices, the market is not closer to theKelly rule than the wealthiest rule. Moreover, there are occasional bursts oflarge deviations from a generally volatile trend toward smaller distances, cf.Table VII.

The dynamics of prices are captured in Figure 5, which depicts one timeseries of 250,000 iterations’ length for the relative price of Asset 1 for each ofthe states s = 1, 2, 3. The benchmark is given by the Kelly prices p∗

1(1) = 0.7,

Figure 4. Euclidean distance between the Kelly rule and the wealthiest strategy resp. themarket prices (which are virtually identical here) for price-dependent strategies.

Figure 5. Relative price of Asset 1 by state and iteration (in 1,000s) for price-dependentstrategies and corresponding Kelly benchmark (horizontal lines).


p∗1(2) = 0.2 and p∗

1(3) = 0.1 in the respective states, cf. Table III. Figure 5shows that prices have a clear tendency to vary around their respective Kellybenchmark levels, except for a short initial period of 2,000 iterations, duringwhich significant mispricing occurs. A comparison with the correspondingcase without price information (Figure 2) reveals that the availability of priceinformation creates a considerable amount of persistent price volatility.

In order to obtain an understanding of the market dynamics with price-dependent strategies, a careful analysis of the successful investment strategiesand their market shares is necessary. It turns out that good performingstrategies use price information in a sophisticated way while, at the sametime, being composed of similar ‘‘building blocks.’’ In particular, usage of themarket portfolio is wide-spread. Averaging across all 250,000 iterations, 30%of the population (representing about 32% of the total wealth) buy the marketportfolio. The strategies which use some price information, without necessarilybuying the market portfolio, amount to 97.5% of the population and representabout 99.99% of the total wealth on average. In the setting with price-dependentstrategies, it seems that price information is necessary for survival.

Individual strategies can be classified according to their behavioral variationover time. Some rules follow a passive strategy and invest 100% of their wealthin the market portfolio at all times. Others adopt an active strategy by alwaysdeviating from the market portfolio. The third group of rules applies a hybridstrategy: in some periods, they passively buy the market portfolio, while inothers they take some active risk by deviating from the market benchmark.Figure 6 applies this classification to the observed behavior of investmentstrategies. It shows how the proportions of wealth managed by each of thesethree sets of rules varies over time. Note that active and passive strategies mayhave the potential to change their behavior, even if they did not do so in the

Figure 6. Percentage of wealth managed by different types of price-dependent strategies.


past, because prices did not take on those values which might trigger theirlatent behavior. The classification therefore gives upper limits on the numberof active and passive strategies and provides a lower limit for the number ofhybrid strategies. Figure 6 also supplies information on the market share ofstrategies that currently take active positions. This set of strategies consists ofall of the active rules and some of the hybrid rules.

There is considerable variation in the number of hybrid strategies deviatingfrom the market portfolio investment. In the early rounds, active strategiesgain and passive strategies lose in terms of market shares. After approximately100,000 iterations, there is a steep increase in the market share of hybrid strate-gies. This occurs because a few rules, which were previously classified as active,bought the market portfolio for the first time during a short period of severemispricing. This is mainly due to price dynamics because, as discussed in detailbelow, strategies typically have a ‘‘trigger,’’ which induces a switch from passiveto active investment behavior if prices enter a certain region of the price space.

Other large jumps in the distribution of strategy types (at 130,000, 165,000and 200,000) are preceded by periods of unusually high volatility, and largedeviations from the Kelly rule. This typically occurs when some extremelyrisky (e.g., under-diversified) strategy experiences a streak of good luck andbecomes wealthy enough to have some market impact before it eventuallygoes bankrupt. For normal levels of price volatility, the presence of passivestrategies amplifies its market impact and contributes to the volatility. For highlevels of volatility, information about past prices gives imprecise informationabout the current market portfolio. This increases the risk for those strategiesthat rely heavily on price information. The gains and losses incurred maychange the wealth distribution, alter the aggregate behavior and cause relativeprices to settle down at new levels. The sharp decline around period 165,000in the market share of currently active strategies is triggered by a change inrelative price levels, following a volatility shock of the type just described.Figure 6 indicates that the main part of this decline can be attributed to hybridstrategies, which change their mode of behavior from active to passive.

In order to investigate the mechanism which causes this shift in behavior, weprovide a detailed analysis of one specific strategy from the experiment—thewealthiest strategy at the last iteration of the experiment. This strategy,denoted λLW(s, p), is present in the population for more than the last 150,000iterations. Because the portfolio weights of λLW are a function of the state s

and the vector of relative prices p = (p1, p2, p3), the rule is therefore of thehybrid type. In states 1 and 2, behavior remains passive throughout, but instate 3, it switches between active and passive modes. Figure 7 summarizes itsbehavior in state 3. In the remainder of this section, we fix s = 3 and refer toλLW(s, p) and λ∗(s) as λLW(p) and λ∗, respectively.

Figure 7 is a contour plot of the distance between the market portfolio andthe portfolio weights of strategy λLW in state 3. Market portfolios, which


Figure 7. Contour plot of δ(p) := ‖λLW (p) − p‖ for relative prices p = (p1, p2, p3) inthe unit simplex. δ(p) is the Euclidean distance between the market portfolio p and theportfolio weights of strategy λLW in state 3. Darker areas represent larger values of δ(p),and white areas represent those prices for which λLW is in perfect agreement with themarket portfolio, i.e., δ(p) = 0. Kelly prices and Kelly portfolio weights are given byλ∗ = (0.2, 0.1, 0.7).

coincide with vectors of relative prices, are represented as points in the unitsimplex. The point λ∗ = (0.2, 0.1, 0.7) is the Kelly benchmark portfolio instate 3 (see Table III). For each price vector p = (p1, p2, p3), λLW(p) are theportfolio weights of strategy λLW , and δ(p) := ‖λLW(p) − p‖ is the Euclideandistance between these two. In the figure, darker shadings corresponds withhigher values of δ(p).

Strategy λLW is in the passive mode (P ) in a subset of the simplex thatconsists of two parts: (1) the line segment defined by all price vectors p withp2 = λ∗

2 = 0.1 and (2) the white triangular area along the north-east borderof the simplex, which corresponds with low prices for Asset 1. The remainingarea of the simplex constitutes the active mode (A). The switch between activeand passive modes is triggered by low prices of Asset 1. The functional formof strategy λLW in the two modes is given by

λLW(p) ={

(1 − λ∗2 − ε(p2))

(p1

p1+p3,

λ∗2+ε(p2)

1−λ∗2−ε(p2)

,p3

p1+p3

)if p ∈ A;

(p1, p2, p3) if p ∈ P .(9)

The function ε(p2) is convex with values ε(0) = 0.008, ε(λ∗2) = 0.000 and

ε(1) = −0.016.In the active mode, strategy λLW chooses a portfolio weight of Asset 2

that is close to the Kelly benchmark. For p2 �= λ∗2, function ε(·) provides

an enhancement which amounts to small long or short positions in Asset 2according to whether this asset is cheap or expensive relative to its Kelly


benchmark. The deviation from the Kelly weight λ∗2 is of the magnitude 10%.

The remaining wealth is invested in the other two assets according to their priceratio, i.e., in the corresponding market portfolio of Assets 1 and 3. StrategyλLW is discontinuous on the border between active and passive modes (whichmarks the switch to passive investment for low prices of Asset 1), exceptalong the line segment where p2 = λ∗

2. However, Equation (9) reveals that thediscontinuities match up to maintain equal ratios of portfolio weights andmarket prices for Assets 1 and 3.

The long-lived strategy λLW is specialized in the sense that it only takesactive positions in one particular asset and in one particular state; otherwise itis passive and buys the market portfolio. A careful analysis of other wealthyinvestment strategies shows that most of them have exactly the same structure.They exploit mispricing in a nearly identical fashion to the strategy studiedin detail above, though for a different asset and a different state. In othersituations, they are passive and buy the market portfolio.

In our Darwinian model, the tournament process punishes investors withlow wealth by a high probability of deletion. Avoiding severe losses is thereforeimportant for survival, a goal that can be achieved by investing part of one’swealth in the market portfolio. Moreover, the logarithm of strategy λLW ’sreturn has a smaller variance than the Kelly rule for about 80% of the time inthe runs reported in Figures 5 and 6. The genetic recombination process favorsthis type of prudent investment behavior because extinction is more likely forrules that have a high volatility of returns.

Specializing in active investment under particular circumstances, whichwould also make sense from a practical point of view, stems from thegenetic process that distributes trading skills across the population. The basicbuilding blocks, which embody particular specialization, spread throughoutthe population of behavior rules. It is extremely unlikely, however, to havethem all combined in one single rule, due to the random nature of geneticrecombination in the tournament process and the lack of individual learning.Though complete knowledge is embodied in the population, this precludes thebirth of a ‘super trader.’ Even though successful investors are more cautiouswhen strategies are price- rather than just state-dependent, convergence tothe Kelly rule prevails because of the push from the population’s aggregatebehavior towards the equilibrium prediction.

5. Conclusion

In this paper, we studied an evolutionary model of a financial market withshort-lived assets, which comprises the two Darwinian processes of selectionand reproduction. The framework generalizes an evolutionary finance modelby applying a genetic programming approach to the creation of variety andthe dynamics of investment behavior. The well-known Kelly rule serves as the


benchmark for optimal investment and asset prices. Our findings fully confirmthe predictions derived from the analytic results in models that neglect thereproductive process which generates diverse behavior. In all experiments, thelong-run outcome shows the emergence of investment styles and market pricesin line with the Kelly rule. When price information is available in additionto information about the current state, strategies make sophisticated use ofthe data by employing the price information to invest part of their wealth inthe market portfolio. This wide-spread behavior provides protection againstsevere losses and increases the chance of survival of the corresponding geneticmaterial. The combination of this type of investment behavior with small betson the Kelly rule eventually drives market prices to the Kelly benchmark.

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